Heterogeneous Nucleation with Finite Activation Energy and Perfect Wetting: Capillary Theory Versus Experiments with Nanometer Particles, and Extrapolations on the Smallest Detectable Nucleus

Abstract

Analytical expressions are provided for charged and uncharged particles in the perfect-wetting limit of Fletcher's classical heterogeneous nucleation theory with finite activation energy G*. A dimensionless logarithm of the critical supersaturation can be represented as a single universal curve versus a dimensionless nucleus radius for neutral particles, while a uniparametric family of curves is required for charged particles. Comparison with experimental critical supersaturations for nanoparticles and molecular ions shows qualitative agreement, with substantial quantitative disagreement. At increasing sizes the predicted critical diameter augmented by a fixed length ∼1 nm approaches the Kelvin limit. Predicted activation probability curves versus size (supersaturation) are steep, whence stringent experimental definitions of the thermodynamic state and the seed particle are required for meaningful comparison with theory. Activation curves for mobility-selected WO_x particles are broader than predicted, even with a well-defined supersaturation, while those of mobility-purified molecular ions agree qualitatively with theory, even in a turbulent mixing nucleation chamber. Recent work by Winkler (2009) based on the Vienna expansion chamber and mobility-purified molecular ions has anticipated these conclusions. The theory is combined with similar considerations on the onset of homogeneous nucleation, to draw conclusions on the smallest nucleus size that may be activated by heterogeneous nucleation. Water is predicted to be able to detect uncharged nuclei with diameters as small as 0.6 nm. The associated possibility to develop detectors for relatively small single neutral molecules is enticing.

1. INTRODUCTION

Following the dramatic advances introduced by Stolzenburg and McMurry (1991) in their Ultrafine Condensation Particle Counter (commercialized as TSI's UCPC 3025), the best condensation nucleus counters (CNCs) became capable of detecting particles with minimal diameters d_min slightly smaller than 3 nm. A common reference used to estimate lower detectable particle diameter d_min has been the critical Kelvin diameter d_KH (Equation (1)), the diameter at which the capillary model predicts that a neutral nucleus surrounded by a supersaturated vapor will require no activation energy to grow into a large drop.

d_KH is calculated at the critical supersaturation S_H for homogeneous nucleation, where γ and v_o are the surface tension and molecular volume of the liquid, k is Boltzmann's constant and T the absolute temperature. For the butanol-based Laminar Diffusion CNC (LDCNC) of Stolzenburg and McMurry (1991), d_KH ∼ d_min , which tended to suggest that CNCs were fundamentally unsuitable for detection of particles smaller than 3 nm. However, in the first experiments in which a Differential Mobility Analyzer (DMA) of high resolution and transmission was used to introduce pure molecular ions into a turbulent mixing CNC (TMCNC), Seto et al. (1997) found that a variety of ions with mobility diameters of ∼1.5 nm were activated. They used the TMCNC of Okuyama et al. (1984), based on the work of Kogan et al. (1960), operated with vapors of dibutyl phthalate (DBP). Subsequent improvements in this instrument (Gamero 1999) showed that there was no limit to the smallest size of a positively charged cluster that could be fully activated by DBP vapors (Gamero et al. 2000c, 2002). These seemingly surprising findings were in fact easily understood in light of (1) Wilson's (1897, 1927) early proof that very small ions can be activated in water vapor, and (2) Kelvin's capillary model explaining why a vapor nucleates more readily on charged than on neutral particles (Thomson 1969). Substantial effort has been devoted over recent years to identify ideal CNC geometries and vapors as well as to build practical instruments with the widest possible size range for the widest possible class of particulate materials, charged and uncharged. The field is progressing so fast that any effort to review it is bound to be quickly outdated (Vanhanen et al. 2011; Sipilä et al. 2009; Iida et al. 2009, 2008, 2007; Hering et al. 2005; Kim et al. 2004; Sgro et al. 2004; Gamero et al. 2000, 2002). Simultaneously, there have been conceptual advances on the general subject of heterogeneous nucleation, and on some basic facts limiting how small a nanoparticle (charged or uncharged) can be detected. This basic leg of the nucleation problem is evidently complex and also far from mature. Still, some elementary considerations based on classical nucleation theory may be made leading to calculable results that can be compared to experiments. Many variants of the theory have been proposed, including consideration for soluble particles (Köhler 1936), the presence of an adsorption layer of solvent on the particle surface (Henson 2007), etc. The underlying capillary theory for insoluble particles without adsorption barriers though allowing for imperfect wetting was put forward by Fletcher (1958, 1962). However, experimental developments on the activation of nuclei smaller than 3 nm have generally been designed and interpreted by the earlier and simpler models of Kelvin and Thomson, which could not explain the observed critical supersaturation well below the predictions of these models. One possible reason for this behavior has been given by Winkler et al. (2008a), who used Fletcher's theory to interpret their own measurements on heterogeneous condensation on charged and uncharged nuclei with diameters approaching 1 nm. Of particular interest was their extension of earlier measurements, limited to charged particles, to show that 1 nm uncharged particles can also be activated, well below the Kelvin diameter, apparently in excellent agreement with Fletcher's theory. The Kelvin and Thomson models presume that nucleation takes place at the thermodynamic condition where there is no activation energy barrier for droplet growth. However, given a finite nucleation time and a finite activation barrier, there is always a finite nucleation rate, whence critical conditions for nucleation arise below the Kelvin or Thomson limits. In other words, the finite activation energy must be taken into account, and this lowers substantially the onset of nucleation below the Kelvin-Thomson value.

One goal of the present study is to compare several existing heterogeneous nucleation measurements with Fletcher's predictions. This task is still pending except for the data of Winkler et al. (2008a), whose computational approach could of course be used to analyze prior data. However, the general trends of the theory have not yet been represented in the form of simple material independent graphs able to provide ready insights on whether or not a concrete measurement conforms to theory. In particular, we do not know if the excellent agreement between theory and measurements reported by Winkler et al. (2008a) is applicable to other data available, either below 3 nm or above. An issue of particular interest is how fast the Fletcher curves approach the Kelvin and Thomson asymptotes at increasing nucleus size. For instance, does the experimental observation of Winkler et al. (2008a) of a substantial reduction below the Kelvin-Thomson critical supersaturation even above 20 nm fit within Fletcher's theory? In order to easily capture these general trends, and compare available data to theory with minimal algebraic complexity, we have derived close form analytical expressions for the various dimensionless parameters of the classical heterogeneous nucleation problem with perfect wetting. In the case of neutral particles a single universal curve describes the critical supersaturation as a function of nucleus size, with only one additional activation energy parameter affecting the form of the activation curves. An additional parameter is required for charged particles. These expressions show that Fletcher's critical size approaches the Kelvin-Thomson values at diameters considerably smaller than 20 nm. The results from these simple formulae are also compared with existing data on the critical sizes and the activation probability for particles a few nm in diameter. We finally combine capillary theory results for heterogeneous and homogeneous nucleation to provide practical criteria to select optimal solvents to activate the smallest possible nuclei. A partial account of this study has previously been given by Fernandez de la Mora (2011).

Before proceeding, some remarks on the merits of pure analysis versus the numerical approach of Winkler et al. are appropriate. The first thing to note is that Fletcher's expressions are singular in the limit of zero wetting angle. This singularity can be avoided by assigning a small wetting angle, but the solution is then refractory to analytical inversion. For calculations involving zero wetting angle it is clearly preferable to write this limit directly, to obtain regular algebraic expressions. Their multiple roots can then be systematically analyzed, providing unambiguous choices for the correct ones. The process may appear as more complex than obtaining a numerical root; but the numerical method still has to decide which is the proper root. It is far simpler and safer to do this once by analysis than to leave it to a program whose choice of roots will tend to change from one machine to the other. Analysis has the additional advantage of minimizing the number of variables involved, and of providing universal (corresponding-state-like) representations valid for any material, where critical data for different substances can be compared in a single figure. The algebraic equations provided here are far more readily programmed than is the inversion of Fletcher's original equations.

2. THEORETICAL CONSIDERATIONS

We shall for simplicity consider first the case of neutral particles, to then analyze the algebraically more complex case of charged particles. The analysis will be restricted to classical nucleation theory (CNT) of insoluble nuclei with perfect wetting, though the method used is suited to incorporate also Fletcher's theory for imperfect wetting (numerically). The discussion includes precisions on the range of validity of the expressions for the activation energy, missing in Fernandez de la Mora (2011).

2.1. Neutral Particles

The excess free energy G(R,R_o,S) needed to form a perfectly wetting drop of radius R over a pre-existing spherical and neutral nucleus of radius Ro (<R) surrounded by its vapor is given by classical capillary theory as

where the surface energy term 4πR²γ+4πR_o ² (γ_nl −γ_ng ) is simplified using the zero contact angle condition

γ_nl and γ_ng are the surface energies between the nucleus and the liquid or gas, respectively. γ and v_o are the liquid-vapor surface tension and the molecular volume of the liquid. For fixed R_o and fixed supersaturation S (>1), G(R,R_o,S) has a maximum G*(R_o,S) at the Kelvin radius R = R_K :

The growth of a seed of radius R > R_K into a large drop is therefore not activated (proceeds downhill), while that of a nucleus of radius R < R_K requires overcoming the activation energy barrier (4a). For later convenience we eliminate S in favor of R_K to express G*(R_o,S) as:

Note that R is an independent variable in the expression (2) for G(R,R_o,S), which takes the fixed value R = R_K in the expression (5) for G*(R_o,S) [or, equivalently, on G(R_o , R_K )]. The key for all criteria used to determine the critical supersaturation S* as a function of the nucleus radius R_o is to assign a fixed value G* to the function G*(R_o,S):

which, once inverted, yields the desired critical relation between S* and R_o

It is often assumed that heterogeneous nucleation takes place at zero activation energy, when the condition (5b) takes Kelvin's simple form:

Following Fletcher (1962), however, and coherently with nucleation theory, Winkler et al. (2008a) note that, given a finite nucleation time t and a finite activation barrier G*, a finite number of nuclei will be activated. In other words, there is no basis for using the classical assumption G* = 0 in Equations (5a–b) to infer the critical relation (5c). Rather, one needs to determine the appropriate value G* > 0 from nucleation theory. And because R_o < R_K , this critical condition corresponds to a smaller supersaturation at given R_o than the Kelvin curve. The appropriate non-zero value of G* must in fact be determined from the criterion that the time t_res available for nucleation be comparable with the time required for an observable number of seed particles to nucleate. The time-dependent evolution of the concentration N(t) of seed nuclei follows from integrating the nucleation rate equation

where K is a kinetic coefficient given by classical nucleation theory (where an analogous K has units of number/s, while our K has units of 1/s and represents the rate at which a single particle crosses the barrier). If one assumes that the nucleation process consumes so little vapor that the thermodynamic state is preserved throughout, then (6) integrates into

The quantity 1−N/N_o is often referred to as the heterogeneous nucleation (or activation) probability. We shall also refer as activation curves to the representation of this probability as a function of either the supersaturation (at fixed nucleus size) or the nucleus diameter (at fixed supersaturation).

TABLE 1 Parameters for conversion of the experimental data into dimensionless variables

Liquid	T (K)	ρ (g/cc)	γ (dyn/cm)	m (g/mol)	α/lnS	S	p v (torr)	K (1/s)	Δt (s)	C	Li (nm)	c	ϵ
DBP	287	1.052	34.10	278.35	0.0640	1120	3.9 10^−6	6.8 10^4	1	11.5	0.484	0.452	6.44
n-propanol	275	0.818	25.13	60.07	0.3455	3.6	4.00	4.9 10^8	10^−3	13.5	0.558	0.520	20.4

The criterion for criticality can now be established by stating that half of the nuclei initially present be activated over the residence time t_res in the nucleation region. In other words, we substitute N/N _o = 1/2 and t = t_res in (7) to find:

This is indeed the critical condition (5a), complemented with a concrete value for the critical activation energy G* = kT ln(Kt_res /ln2) fixed by nucleation theory. The constant C = ln(Kt_res /ln2) is typically a large number, which, due to its logarithmic variation with K and t_res , is traditionally expected to have little dependence on the circumstances of the device. We shall see, however, that this dependence is not always so small. Fletcher (1962) estimates K ∼10⁷ s⁻¹ for ions in water, leading when t_res ∼ 1 s to C = 16.5. An idea of the ambiguity involved is provided by the different estimate in Fletcher (1958) for neutrals in water, yielding an increase in K by over 5 orders of magnitude (C ∼ 28). Keeping in mind that the ambiguity in C is not trivial, we will for the time being estimate K through

based on the collision rate of vapor molecules with seed particles, where p_vS is the partial pressure of vapor, and m its molecular weight (see Volmer 1979; Winkler et al. 2008a; Fletcher 1962 for more refined expressions). Given the ambiguities in K, the surface area 4πR_K*² of the critical embryo appearing in Equation (9) for the prefactor will be based on the assumption R_K* = 1 nm. No such simplification will be used to compute the exponent. Corresponding values of K and C are shown in Table 1 for relevant experimental data in n-propanol and DBP.

In order to cast the predictions of the theory in its simplest and most general (material property independent) form, we introduce a length L_o to define the dimensionless radii of Equation (10), and the supersaturation parameter β defined in Equation (11). For simplicity we have chosen L_o in Equation (12) such that the critical relation between S and R_o is universal for all materials (see below):

The number N of surviving nuclei at the exit of the CNC (or at the end of the nucleation pulse in an expansion CNC) is given by Equation (7) particularized at t = t_res , which can be re-expressed in terms of C instead of Kt_res [i.e., Equation (8)] as

This takes the following form in terms of the dimensionless variables (10–11):

This equation is restricted to x_o < 1/β. For x_o > 1/β there is no activation energy and N/N_o = exp[−ln2 e ^C ]. Given the large typical values of C and the double exponential, N becomes transcendentally small (full activation). The form of the activation curves (13b) is shown in Figure 1 as a function of nucleus size for several fixed values of the parameter C and the supersaturation variable β. The rise taking place near a critical radius x_i* can be seen to be rather sharp, particularly at the largest values of C. Critical conditions are defined by the criterion N/N_o = 1/2, which takes place independently of C when the second exponent in (13b) vanishes:

The critical condition (14) fixes a relation β = β*(x_o ) between the seed radius x_o and the critical supersaturation variable β*. It is noteworthy that this critical curve is universal for all vapors, as it contains no material dependent parameters. If one were to allow for a finite wetting angle a family of curves would result.

FIG. 1 Predicted activation characteristics of neutral particles at 9 different super-saturations (β = 0.44213; 0.40466; 0.37160; 0.34271; 0.31750; 0.295435; 0.27604; 0.25890; 0.24367, corresponding to x_o* varying from 1 to 3 in 0.25 intervals), each at 4 values of C (10, 16, 40, 100, with slopes increasing with C). Reproduced from Fernandez de la Mora (2011).

FIG. 2 Dimensionless critical supersaturation versus dimensionless seed nucleus diameter for neutral particles, showing the exact prediction Equation (15) (thick line peaking at x_o = 0), and three approximations x_o ⁻¹, (x_o +1)⁻¹ and Eq. (21) (lowest line going through a maximum). Reproduced from Figure 32.7a of Fernandez de la Mora (2011).

The critical relation is a cubic equation for both β* and x_o , whose complicated relevant root may be simplified into:

This curve is shown in Figure 2. It can also be constructed parametrically in terms of the variable ξ = β*x_o , by rewriting Equation (15) as

and using the identity

The Kelvin limit β* = 1/x_o corresponds to ξ→1. In this region of almost zero activation barrier, G* is (by construction) quadratic in 1/β* −x_o , hence in 1 −ξ. Equation (15) can indeed be written as

and expanded into the useful form:

To first order in (1−ξ) one finds the approximate explicit relation

As shown in Figure 2, this approximates the exact behavior β*(x_o ) much better than the Kelvin limit β* = 1/x_o . Within this approximation, the Kelvin limit holds as long as the particle radius is substituted by (R_o + L_o ). A horizontal displacement of the Kelvin curve to the left should therefore yield the Fletcher curve to a first approximation. This prediction is approximately met by the numerical calculations of Pichelstorfer et al. (2010). The criterion of approximate validity of the pure Kelvin limit is that x_o >>1. Carrying the expansion to higher order in powers of (1 + x_o)⁻¹ provides the following description, also shown in Figure 2:

It is noteworthy that a finite critical supersaturation β* = 1/√3 is predicted even for x_o = 0. We shall see, however, that arbitrarily small neutral particles cannot generally be activated because homogeneous nucleation sets in typically at β* < 1/√3 (section 4). In the vicinity of the critical condition x = x_o *(β), the exponent in Equation (13b) can be linearized into

where for brevity we write x_o* instead of x_o*(β). Equation (22) is an excellent approximation as long as x_o* >1, C >10. From Equation (22), the relative width for N/N_o to rise from 20% to 80% is

This width determines the sizing resolution achievable by heterogeneous nucleation of neutral particles. The group x_o [1-β x_o ] in (23) tends to 1 at large x_o (x_o > 10), where the resolution may be quite high, tending to Cx_o (94 at x_o = 10, C = 10). It decreases down to 0.91C at x _o = 1.4.

2.2. Charged particles

The generalization of Equation (2) including a Coulombic term is

where ϵ_o and ϵ are the electrical permittivity of vacuum and the dielectric constant of the drop. Because new dimensionless variables need now to be defined, we use R_i (i for ion) rather than R_o for the nucleus radius. We use the new characteristic length L_i (different from L_o ) based on the Rayleigh length Equation (25a), together with the new dimensionless supersaturation variable α:

The maximum in G is given by the condition

which fixes y as a function of α through the inverse function form

Equation (27) generalizes the condition R = R_K (x = 1/β) previously found for neutral particles. The dependence α(y) is non-monotonic, with a maximum

There are no real roots y(α) for α > α*. For α < α* there are two real roots, y⁺ (α) and y⁻ (α), each uniquely defined by Equation (27) and by the ordering criterion

For α<α*, G(y) has a minimum at y = y⁺ (α), and a maximum at y = y⁺ (α), so the proper choice of y for the computation of the activation barrier is G*(x_i,α) = G[y⁺ (α),x_i,α], or:

It is important to note that the validity of Equation (30) is restricted as follows:

The domain (31a) has zero activation energy and corresponds to Thomson's classical region of nucleation. The region x_i < y⁻ (α) corresponds to seed particles smaller than the value of x_i at which G is a minimum. These small particles have a large Coulombic energy and tend to grow naturally (energetically downhill) by picking up vapor until occupying the energy minimum with a dimensionless radius y⁻ (α). This is why the activation energy in this region takes the same value as at x_i = y⁻ (α) (Figure 3).

FIG. 3 Dimensionless activation energy versus ion size for α = 0.2461, 0.2790, 0.32099, 0.3744, 0.4375 (top to bottom), showing a constant value below x_i = y⁺ (α).

Equation (30) leads to the following expression for the activation curve (13a)

where we have defined the new dimensionless parameter c:

The same interpretation of y as y⁺ (α) applies to Equation (32) as to Equation (30), with the same restrictions (31) on the domain of validity. In the range (31a) where G = 0, the exponent in Equation (32) is C, and N/N_o = exp[–ln2 e ^C ]. Given the large typical values of C and the double exponential, this implies full activation (N = 0).

The two constants C and c enter now explicitly in the expression (32) for the activation curves. The ratio C/c can be seen in Equation (33) to be substance-dependent, but independent of the somewhat ambiguous parameter Kt_res . Figure 4 shows the predicted activation curves for the special case C = 15, c = 0.45, while fixing the supersaturation parameter α and varying x_i (left), or vice-versa (right). The curves can be plotted directly as a function of x_i , and parametrically as a function of α = 1/y–1/y ⁴ (taking the precaution of selecting the parameter y > 4^1/3). At high supersaturations one can see on the left figure that even the smallest ions have a substantial probability of activation (non-zero horizontal asymptote at small x_i ). This size-independent behavior of the smallest ions is repeated on the right curve, where ions smaller than the ones corresponding to the curve farthest to the right fall on top of the same curve.

The critical supersaturation associated to the condition N/N_o = 1/2 is again obtained by cancelling the exponent in Equation (32):

FIG. 4 Activation curves for ions, with C = 15 and c = 0.45. Top: varying ion diameter at fixed supersaturation: α = 0.3804, 0.3744, 0.3627, 0.3409, 0.3210, 0.3030, 0.2866, 0.2718, 0.2584, 0.2461 (from left to right). The two leftmost curves continue horizontally at decreasing x _o. Bottom: varying α and fixed ion diameter: x_i = 1.25, 1.5, 1.75, 2, 2.25, 2.5, 2.75, 3, 3.25, 3.5, 3.75, 4, 4.25, 4.5, 4.75, 5 (from right to left).

which fixes the critical relation between x_i and α. Equation (34) is a 4th order polynomial in x_i , giving x_i(y) parametrically. Its physically meaningful root is the smallest of the two branches x^± [y] in Equation (35) resulting from sign choice in ±p[y]. The + sign applies everywhere, except at values of c >1.156, when p[y] changes sign at a value y = y_o within the physically meaningful range. The branches –p[y] and +p[y] then apply at y < y_o and y > y_o , respectively. The sign selection in r^± is irrelevant, as only the sum (r⁺+r⁻) appears in Equation (37)

Predicted critical curves α*(x_i) are shown in black in Figure 5 for the indicated values of the activation energy parameter c defined in Equation (33). The limit c = 0 gives the especially significant Thomson curve. Only its piece to the right of the maximum α⁺ (x_i ) is physically relevant for nucleation. The branch α⁻(x_i ) to the left of the maximum corresponds to a minimum rather than a maximum in G(y), below which ions are solvated and above which they are dry (as long as they do not nucleate). The activation curves for c > 0 terminate to the left at their intersection with the Thomson curve. As a result of solvation, all initially smaller ions behave at given α exactly as the ion with the equilibrium solvation size. The curves for neutral and charged particles merge approximately at x_i > 4, with a slight dependence on c.

FIG. 5 Critical supersaturation of charged and neutral particles. Black continuous lines are for charged particles, showing predictions from Equation (35) at several values of the activation energy parameter c (1.5; 1; 0.5; 0.25; 0, from bottom to top). Dashed theoretical lines are for neutral particles. Thomson's curve, α* = 1/x_i −1/x_i ⁴ is shown in thicker trace. Its rising piece marks the boundary between dry (right) and solvated (left) ions; its decaying piece coincides with the curve for c = 0. Black data points (Gamero et al., 2002) are for cluster ions in DBP. The grey data points and the two thick grey segments near them are the data and theoretical calculations from Winkler et al. (2008a). After Figure 32.8a of Fernandez de la Mora (2011).

3. COMPARISON BETWEEN THEORY AND EXPERIMENTS

Prior to the work of Winkler et al. (2008a), the Thomson curve had been used as theoretical reference for comparison with the few size-resolved data available in this size range, all corresponding to charged particles (Gamero et al. 2000, 2002). For unexplained reasons their data for x_i >1 fell substantially below the Thomson curve. The same phenomenon was reported by Winkler (2004) for considerably larger singly charged particles with diameters going up to 24 nm. Winkler et al. (2008a,b) have provided new data for small charged and uncharged tungsten oxide (WO_x) particles in n-propanol vapor, also showing activation well below the Thomson and the Kelvin curves. They provide the first explanation for this peculiarity in terms of Fletcher's theory for heterogeneous nucleation. Fletcher includes two effects: one associated to the finite value of G*, the other due to imperfect wetting of the nucleus by the vapor. Winkler et al. (2008a) consider both, but in the end assume almost perfect wetting, so their calculations probably account simply for the finite activation energy. Figure 2 of Winkler et al. (2008a) shows excellent agreement between theory and experiments for all the size ranges studied, and for charged and neutral particles. The calculation is strictly numerical, without details on the selection of branches. It also includes all the algebraic complexity of the imperfect wetting theory, which is singular near the limit of perfect wetting. Figure 5 collects both the earlier charged cluster data (black) and the new neutral WO_x data (grey). The charged WO_x data are not shown, as those for positive polarity fall on top of the neutral particle data, and those for negative charges are only slightly below. In order to permit comparison between charged and uncharged particles, the theory and the data for uncharged particles are made dimensionless with the same characteristic length L_i used for ions. This turns Equation (14) into

which reintroduces the dependence on the parameter c that was removed in the representation of Figure 2. The theoretical curves hence depend on the value of the activation energy parameter c for both, charged (continuous lines) and neutral particles (dashed lines). One sees that the ion cluster data (black) fall close to the black curve for c = 0.25 at intermediate sizes, but are above it for small and large particles. Of particular interest is the rapid transition towards the Thomson asymptote α* = 3/4^4/3 seen at about the intersection between the data and the α⁻ (x_i) curve. The estimated value of c shown in Table 1 is 0.45. Hence, although the effect of the finite activation energy does not lead to agreement between theory and experiments, it has the virtue of bringing all the data above the theoretical curve. The discrepancy with theory could then conceivably be due to imperfect wetting, which always increases the energy barrier for nucleation. Another possible explanation may be due to higher solubility of the large clusters in DBP (made up of hydrophobic tetraheptylammonium bromide) versus the smaller and increasingly hydrophilic tetra-alkylammonium salts used to produce the smaller ions. The WO_x data for neutral particles fall close to the grey theoretical curve associated to c = 1.5 for x_i < 2.5, but are substantially below for larger diameters. This is clear in the figure for the two larger grey data points shown. The data for even larger WO_x nuclei (previously reported by Winkler 2004) fall increasingly below the theoretical curve. There are also some differences between the theoretical curves of Winkler et al. (2008a) (taken from their Figure 2 and shown with thick grey lines in Figure 5) and our own. Their curve for neutral; particles (dashed thick grey line) falls very close to our own curve for c = 1.3 (not shown) at x_i < 2.3. However, it deviates downward at larger diameters, approaching artificially their experimental point at x_i ∼ 3.3. Their calculated curve for charged particles presents a similar anomalous downward bend at the largest particle size. It is also insufficiently curved at the lowest sizes, and crosses incorrectly the Thomson curve without having become horizontal. These differences in the two theoretical calculations are nonetheless modest at the scale of the disagreement between theory and experimental points. Note also that our estimated c for these experiments is 0.52 based on Equation (9), corresponding to Kt_res over 11 orders of magnitude below the value required to yield c = 1.5, This difference is probably partly due to our rudimentary model for K. A peculiarity of the data of Winkler (2004) for particle diameters even above 20 nm is that the experimental critical supersaturation remains well below the Kelvin value both for nonane (Winkler et al. 2008c) and propanol (Winkler et al. 2008a; supplementary information), while CNT predicts an earlier convergence.

In conclusion, the main discrepancy is not between the two theoretical curves, but between the theory and both sets of data. Particularly noteworthy is the lack of convergence of the data of Winkler et al. (2008a–c) at large particle sizes on the Fletcher or Kelvin curves (which in this region are simply slightly horizontally displaced from each other). Plotting these data as 1/α* versus x_i yields a straight line for the larger particles, as expected. But its slope is 1/0.63, as if the group 2γv_o /(kT) was a factor 0.63 below what is expected for n-propanol. A similar linear relation between lnS* and 1/d_p with a non-Kelvin slope can be seen in Figure 5 of Winkler et al. (2008c) for nonane vapors. Chen and Cheng (2007) have previously observed an analogous need to decrease the surface tension to fit their own data to the Kelvin curve, also at relatively large diameters (10–15 nm). Several prior studies with large particles have compared critical diameters for heterogeneous nucleation in water vapor with the corresponding Kelvin diameter, and found either agreement (i.e., Liu et al. 1984 for dioctyl phthalate drops 18 and 56 nm in diameter in water vapor), or diameters larger than Kelvin's (i.e., Porstendorfer et al. 1985 for Ag particles 6–19 nm in diameter, and Alofs et al. 1995 for silver particles 16, 19, and 30 nm in diameter).

Another useful test of the capillary theory can be made by comparing it with measurements for the full activation curve rather than just the point at N/N_o = 1/2. Such comparisons often show considerably slower rises of the curve for the experiments than for the theory. This is not surprising because the predicted activation curves are quite sharp both upon varying the supersaturation (α) or the particle diameter (x_i ). Therefore, no theoretical information can be drawn from the width of the experimental curves unless the experiment is carried with a sharply defined supersaturation, and with a highly monodisperse aerosol that must in addition be of uniform composition and shape. As a reference, taking c = 0.5, C = 15, near the condition when the ion becomes dry, (x_i ∼ 1.22), α varies only by 1.9% as the activation probability rises from 20% to 80%. It is therefore apparent that ln(S) must be fixed experimentally within better than 2% for the experimental width of the activation curves to be theoretically relevant. It is presently unclear whether this sharp definition is possible with existing instrumentation. Probably the best devices from the point of view of fixing precisely the supersaturation in space and time are isentropic expansion CNCs. Although there are hot boundary layers near the walls, these can be avoided by restricting the optical detection to particles near the center of the nucleation region. Even so, most of the data obtained by even the most refined among such instruments have exhibited considerably wider activation curves than predicted here. This may be seen in the small gray data points shown in Figure 6 for negatively charged tungsten oxide nanoparticles in n-propanol (taken from Figure 1 of Winkler et al. 2008a). Besides dispersion in the experimental S, other possible causes for this mismatch are lack of uniformity in ion mobility (associated to the finite resolving power of the DMA used for size selection), chemical composition (oxidation of tungsten even under the cleanest conditions produces a complex mixture of different oxides), charging ion (many different ions are produced in radioactive chargers, particularly those not using ultraclean standards), and particle shape.

FIG. 6 Comparison of predicted and measured activation curves for charged particles: small gray (cluster ions from Winkler et al. 2011) and large gray data (negative WO_x ions from Winkler et al. 2008a) are in n-propanol. Black data (THA_n+1Br_n ⁺ with n = 2, 5, 8, 10, 12, right to left) are in DBP. Continuous lines are from Equation (32), with x_i selected to match the data at N/No = 1/2, (c, C/c) taken to be (0.25, 25.9) for 2-propanol and (0.2, 25.4) for DBP.

These inhomogeneities are difficult to avoid by conventional aerosol techniques, but they may be overcome by using DMA-purified molecular ions from electrosprayed solutions (Fenn et al. 1989; Rosell et al. 1996; Seto et al. 1997). Figure 6 shows activation data in black for THA_n+1Br_n ⁺ clusters (n = 2, 5, 8, 10, 12) in DBP vapor, inferred from figure 6 of Gamero et al. 2000. The calculations are for c = 0.2 (a factor over two below the estimated experimental value, as a larger c would preclude the observed dependence of critical supersaturation on ion size) and for C/c = 25.4 (appropriate for DBP), with x_i chosen, irrespective of its experimental value, to match the experimental α _1/2. Interestingly, the observed activation curves are almost as sharp as those predicted. This is unexpected for a turbulent mixing CNC, where unit supersaturation at the walls coexists with S ∼ 1000 at the core. Some nonidealities are observed in the form of tails at the low α end of the distributions. This mismatch is almost surely due to the selection by the DMA of a small fraction of multiply charged ions with the same mobility as the main singly charged cluster. This problem is to be expected from the shoulders seen in the mobility distribution of the partially discharged aerosol from which the test ions were selected (see figure 4 of Gamero et al. 2000). Because multiply charged ions are activated at reduced α, the sharp activation curves seen on the right side of the black curves confirms the fact that the supersaturation is indeed very well defined in this mixing instrument. This circumstance provides a first indication that the broad activation curves for WO_x just discussed are due to lack of homogeneity in the nuclei rather than imperfect definition of S. This point is confirmed more directly by recent very narrow activation curves for ion clusters in n-propanol (black data points in Figure 6), taken in the same Vienna expansion CNC used for the broad WO_x measurements. I am much indebted to Prof. Paul Wagner and his colleagues (Winkler et al. 2011) for kindly letting me use these still unpublished data. Winkler (2009) has previously noted the sharp rise of these data, and their eminent suitability to provide unique theoretical information. This singular finding provided our motivation to compare Gamero's data to theory.

We conclude that, although the experimental relation α _1/2(x_i ) does not match the capillary theory in either DBP or n-propanol, the narrow widths and the general shape of the theoretical activation curves do generally match the theory. This trend provides qualitative confirmation of the fact that the supersaturation in the Vienna expansion CNC and in the turbulent mixing CNC of Gamero et al. is defined sharply enough to enable theoretical inferences based on the shape of the activation curve. Note however that the steepness of the curves depends on the choice of c, and doubling c leads in both data sets to a clearly sharper theoretical versus experimental curves. Recent work by Vanhanen et al. (2011) on a turbulent mixing CNC (the first commercial CNC sensitive to charged particles down to 1 nm) also reports relatively sharp activation curves for molecular ions.

TABLE 2 Characteristics of various working fluids relating to the smallest detectable size

Liquid	T (K)	γ (dyn/cm)	Li (nm)	2.7√(kT/γ) (nm)	kT/(γLi ^2)	CH-max
DBP	287	34.10	0.484	0.913	0.495	37.9
n-propanol	275	25.13	0.558	1.041	0.485	38.6
acetonitrile	277	31.32	0.522	0.936	0.448	41. 9
n-nonane	293	22.88	0.462	1.126	0.827	22. 7
water	267	76.50	0.390	0.588	0.317	59. 2
methanol	284	23.24	0.576	1.100	0.508	36.9

Laminar diffusion CNCs normally have spatially dependent supersaturations, and would tend to generate a thermodynamic state less homogeneous than a turbulent mixing CNC. Nonetheless, one cannot preclude the possibility that, once tested with molecular ions, some laminar diffusion CNCs might also yield sharp activation curves. Hopefully, use of clean chargers and improved aerosol generation techniques will also provide sufficiently sharp activation curves for a wider class of aerosols than the molecular ions and ion clusters discussed here.

4. FLUID SELECTION CRITERIA BASED ON CAPILLARY MODELS

4.1. Homogeneous Nucleation

Homogeneous nucleation eventually takes place at a critical supersaturation S_H at which drops of vapor form without the need for seed particles. The smallest nucleus that can be detected by vapor condensation must therefore be activated at a supersaturation S* < S_H . Since both S* and S_H depend on fluid properties, selection of an ideal fluid for small particle detection requires an ability to predict S_H . The critical activation energy G_H* for homogeneous nucleation is fixed similarly as before, now by the condition that fewer nuclei form homogeneously than heterogeneously, such as not to obstruct the observation of heterogeneously formed drops. The homogeneous nucleation rate n_vK_H exp[−G_H*/kT] must then be smaller than, say, N/t_res , where n_v is the number concentration of vapor. We shall for simplicity ignore the differences between K and K_H , since the effective collision diameter for a vapor molecules with a typical homogeneously nucleating cluster is quite comparable to the 1 nm assumed for the calculation of K. Associated errors are comparable to those introduced in the calculation of K, and affect the results only through small logarithmic terms (see Volmer 1979 for more refined expressions). Hence, ignoring the reduction of n_v during nucleation, all the reasoning followed before for heterogeneous nucleation on neutral particles remains valid here, with the simple substitution of C_H instead of C, and the selection of x_o = 0 turning (14) into (42):

4.2. Heterogeneous Nucleation

We shall consider here only the more difficult case of neutral particles, previously discussed in less accessible form by Fernandez de la Mora (2011). This earlier study included also the simpler case of charged particles, which are much easier to activate (even at subnanometer sizes), apparently without the need for exceptional working fluids. The criterion (42) can be expressed as

β*_max is always smaller than 1/ √3 because, given that n_v>>N under all circumstances relevant to heterogeneous nucleation, C_H = C + ln(n_vln2/N) is always larger than C. The limitation β* < β*_max amounts to forbidding access to the region between the maximum of the critical curve (thick line, β* = 1/√3) in Figure 2 and a factor (C/C_H)^1/2 below it. Consequently, arbitrarily small neutral nuclei cannot be activated. For instance, in an extreme situation of a very volatile vapor and a very dilute aerosol (n_v = 10¹⁹ cm⁻³, N = 1 cm⁻³), ln(n_vln2/N) = 43.4. More realistically, in the work of Gamero et al. (2002) based on electrospray generated clusters, at a typical sample flow rate of 1 lit/min, their electrometer was measuring 100–1000 fA of mobility-selected ions, corresponding to N varying from 3.75 10⁷ to 3.75 10⁸ cm⁻³. This may be considered a high concentration for a monodisperse aerosol of 1 nm particles. For their DBP data n_v ∼ 10¹⁴ cm⁻³, C = 11.5, leading to C_H/C between 2.16 and 2.36, with √3β*_max = (2.26)^−1/2 ∼ 2/3. This corresponds in Figure 2 to (x_i)_min ∼ 1.4. The Kelvin criterion would give the considerably larger value (x_i)_min = 2.6. In either case, the smallest neutral particle that can be activated is limited by homogeneous nucleation to a radius R_min = (x_o)_min(CkT/4πγ)^1/2 . In our estimate, the factor (x_o)_min(C/4π)^1/2 is 1.34:

As shown in Table 2, this corresponds to a minimum detectable particle diameter between 0.9 and 1.1 nm for all the organic solvents listed (including n-propanol, in qualitative agreement with the findings of Winkler et al. 2008a), and to 0.6 nm for water! When based on the Kelvin diameter, the 1.34 coefficient of Equation (28) would become 2.5, almost twice as large. A proper perspective on the almost incredibly small size given by Equation (44) may be obtained by comparing it with the result of Magnusson et al. (2003), perhaps the most reliable information available until 2008. These authors give the correlation (for CNT at 300 K) d_KH (nm) = 17.3 γ^−1/2 + 0.37 (γ in dyn/cm). Ignoring the small 0.37 nm shift, this corresponds to R_min = 4.25(kT/γ)^1/2 . The large difference between their coefficient 4.25 and our value 2.5 for the Kelvin diameter is due to the much larger C_H typical of homogeneous nucleation problems, where measurable nucleation rates of 1/cm³/s are possible. The fact that a less stringent criterion applies under favorable conditions for homogeneous nucleation in competition with heterogeneous nucleation drops their 4.25 factor almost by one half. The finite activation energy and the lack of applicability of the Kelvin criterion at ∼1 nm stressed by Winkler et al. (2008a) lower this coefficient further by almost twofold. The overall effect is dramatic. Based on Magnusson et al. the smallest neutral particle one could expect to detect with a singular liquid having surface tension as high as water had a diameter d_min = 8.5(kT/γ)^1/2 = 1.85 nm. Now theory tells us that, under favorable conditions, this diameter is only 2.7(kT/γ) ^1/2, namely, 0.59 nm for water. This may seem implausible; however the corresponding prediction of 1.05 nm for n-propanol has already been approximately confirmed experimentally by Winkler et al. (2008a). The associated possibility to develop detectors for moderately large single neutral molecules such as the explosive RDX is enticing. In contrast to alternative condensation methods for single vapor molecule detection (Kogan 1998; Van Luik and Rippere 1962), this detection does not require prior physical or chemical transformation of the vapor molecule.

In conclusion, the smallest neutral particle diameter detectable depends principally on (kT/γ)^1/2 , favoring low T and high γ. There are other smaller but useful effects of other parameters through the quantities C and C_H which may provide advantages for some fluids and need to be considered in each specific situation. Iida et al. (2009) have suggested that, besides γ, another key parameter favoring low R_min is a low vapor pressure, which led them to study oleic acid and dioctyl sebacate (DOS) (d_KH ∼ 1.87 nm for both). The role of vapor pressure is clear in C = ln(t_resK/ln2), since K ∼ n_v . It is even more important in C_H = ln(Kt_resn_v /N) ∼ ln(t_resn_v ²/N). Since one seeks to reduce C_H , it is plain that besides the benefits already described of increasing N, it is similarly useful to decrease t _res, and twice as advantageous to reduce n_v . These considerations confirm the views of Iida et al. (2009) about the interest of low volatility fluids.

5. CONCLUSIONS

Based on capillary theory for heterogeneous nucleation, closed form solutions have been obtained for the activation probability and the critical size of spherical perfectly wetting seed particles, charged and uncharged. As previously observed, the predicted critical nucleus is below the Kelvin-Thomson limit for very small nuclei. It approaches the Kelvin diameter at nucleus sizes large compared to a characteristic length L_o = (CkT/4πγ)^1/2, where C ∼ 10. The activation curves are very steep, both upon varying the nucleus size and the supersaturation. Highly homogeneous nuclei and supersaturations are therefore required in meaningful measurements of the structure of activation probability curves. As expected, isentropic expansion devices are capable of achieving this sharp supersaturation requirement. Surprisingly, so is the turbulent mixing instrument of Gamero et al. (2000). In contrast, no seed particle produced by conventional aerosol generators has yet achieved the necessary uniformity in the seed nucleus, with the exception of mobility classified molecular ions produced by electrospray ionization of controlled solution ions. Theoretical considerations on the critical supersaturation for homogeneous nucleation are made, similarly to those previously made on heterogeneous nucleation. From these we draw criteria on the smallest neutral cluster that can be activated with the most favorable working fluid. This turns out to be water (or a high surface tension liquid) at low temperature, for which a smallest detectable diameter of 0.6 nm is expected. This implies the real possibility to detect single vapor molecules by condensation.

Acknowledgments

I thank M. Attoui, P.H. McMurry, S.V. Hering, M. Kulmala, and P. Winkler for a number of very useful discussions, and P. Wagner for many insights and for his critical reading of the manuscript. I am grateful to my SEADM colleagues for their support of this work, and thank particularly Guillermo Vidal and Roberto Alonso for sharing their ideas on the development of particle detectors. This article draws material from Fernandez de la Mora (copyright John Wiley & Sons, 2011), parts of which are reprinted with permission of John Wiley & Sons, Inc.