Unraveling the"Pressure-Effect"in Nucleation

The influence of the pressure of a chemically inert carrier-gas on the nucleation rate is one of the biggest puzzles in the research of gas-liquid nucleation. Different experiments can show a positive effect, a negative effect, or no effect at all. The same experiment may show both trends for the same substance depending on temperature, or for different substances at the same temperature. We show how this ambiguous effect naturally arises from the competition of two contributions: nonisothermal effects and pressure-volume work. Our model clarifies seemingly contradictory experimental results and quantifies the variation of the nucleation ability of a substance in the presence of an ambient gas. Our findings are corroborated by results from molecular dynamics simulation and might have important implications since nucleation in experiments, technical applications, and nature practically always occurs in the presence of an ambient gas.

A gas phase can be supersaturated considerably beyond its equilibrium condensation point before liquid drops form spontaneously. The pathway of the phase transition is blocked because microscopic droplets are thermodynamically less favorable than the bulk vapor. Therefore, the transition can only be initiated by rare fluctuations exceeding a critical size, called the critical nucleus. The formation of such a critical nucleus is the limiting step in the transition and its frequency of occurence is called the nucleation rate. Nucleation is behind most phase transition and plays a crucial role in atmospheric processes such as the formation of aerosols or the condensation of water vapor into clouds [1,2]. An accurate experimental evaluation of the nucleation of atmospherically relevant substances and its correct theoretical prediction are essential for a better understanding of climate change and are the subject of intense investigations [3].
Nucleation is highly sensitive to small changes in the state variables describing the system, most notably to temperature. However, condensation is inevitably connected with the release of latent heat. In experiments this latent heat is removed by the presence of a large background of carrier-gas. This carrier-gas should be noncondensing and chemically inert and should have no influence on the nucleation except serving as the desired heat bath. Nevertheless, many experimental results suggest that there can be an influence on the nucleation rate that can span some orders of magnitude, depending on the pressure and type of the carrier-gas (see Refs. [4,5] and references therein). Unfortunately, the overall picture is far from clear. Comparisons of experimental findings cover all possibilities: no effect, increase, or decrease of the nucleation rate with carrier-gas pressure. There were some concerns if the pressure-effect is not just an experimental artifact, which cannot be completely ruled out in all cases. However, continuous improvements in the experimental setups rather confirmed than remedied this undesireable and elusive effect.
Following the experimental evidence, theoretical works have tried to explain the pressure-effect with varying suc-cess. These approaches include changes to classical nucleation theory (CNT) accounting for nonideal behavior of the vapor and/or carrier-gas [6], treating the carrier-gas as not fully inert or the problem as binary nucleation [7,8,9] or analyzing its influence on cluster stability and the impingment rate [10], and many more. There remain many contradictions and differences both in the direction and in the magnitude of the effect. These are complicated by the discrepancies between experimental results themselves and in the comparison of experiment and theory.
Here we present a simple yet physically very appealing model that resolves many of the apparent contradictions of the pressure-effect. We take a deliberate step back and incorporate the presence of a carrier-gas into classical theory in a most natural manner that accounts for the two primal contributions of the carrier-gas: the efficiency of thermalization and the additional work that a cluster has to spend for growing in its presence. These contributions have opposite trends and we show how this may be responsible for the existence of apparently contradictory results.
Nucleation theory [11] usually aims at calculating isothermal nucleation rates, taking the constant value of the temperature and the idea of a non-influential carrier-gas for granted. In CNT, the work of formation of a droplet of size n at constant pressure p and temperature T is [11] ∆G (n) = −n∆µ + s 1 γn 2/3 , i.e., the combination a volume term related to the difference in chemical potential ∆µ between the vapor and the liquid and a surface term needed to build the liquid interface A = s 1 n 2/3 with the surface tension γ (s 1 = 36πv 2 l 1/3 is the surface area per monomer and v l is the volume per molecule in the bulk liquid). The vapor pressure p is assumed as ideal and the liquid cluster is considered as an incompressible spherical drop with a sharp interface and bulk liquid properties. The free energy has a maximum at the critical size n * and its height ∆G * is The isothermal steady-state nucleation rate then is where K is a kinetic prefactor. In CNT, the difference in the chemical potential is given by Here, S = p/p eq is the supersaturation and p eq is the equilibrium vapor pressure. The second term in Eq. (4) is arising from the pressure-volume work the liquid drop has to perform against the ambient vapor pressure [13]. It is typically small and hence commonly neglected but we keep it here for clarity. We now naturally account for the presence of an ideal carrier-gas by noting that the cluster must also perform pV -work against the ambient carrier-gas pressure, W c = p c V (n) = nv l p c [14]. The work of formation including pV -work now reads where we have cast the pV -contributions into an "effective chemical potential" Replacing ∆µ for ∆µ eff in Eq. (2), the barrier becomes ∆G * pV = (16π/3) v 2 l γ 3 /∆µ eff 2 and the pV -corrected rate follows directly from Eq. (3) as The main designated role of a carrier-gas is to keep T constant. However, this thermalization might not be perfect and nucleation then happens under nonisothermal conditions. The classical work of Feder et al. offers analytical expressions to quantify this influence on the nucleation rate [15,16]. Physically, it is controlled by the competition between the energy increase due to latent heat and the energy removal through elastic collisions with vapor and carrier-gas molecules. The parameter quantifies the energy released when a monomer is added to a cluster, which is the latent heat h per molecule (corrected by a small amount k B T /2) minus the energy spent on increasing the surface area A(n) against the surface tension. The mean squared energy fluctuation removed by collisions with impinging molecules is Figure 1: Deviation of the nucleation rate from CNT due to the full pressure-effect, Eq. (12), as a function of the ratio of carrier-gas over vapor molecules (solid) arising from the two contributions of nonisothermal effects (dash, Eq. (10)) and pV -work (dash-dot, Eq. (7)) for Argon at 50 K and S = 869.
for an ideal monatomic vapor and carrier-gas [16] (m is the molecular mass, N the number of molecules of the condensable). Eq. (9) indicates that a large number of carrier-gas molecules N c and a light (small molecular mass m c ) carriergas are most effective for a good and fast thermalization. Finally, the influence of nonisothermal effects on the steadystate nucleation rate is determined by the combination of q and b: We now obtain the full pressure-effect (PE) of the carriergas on the nucleation rate by combining Eqs. (10) and (7), taking the latter as the isothermal but pV -corrected rate: Obviously, the magnitude of the rate calculated by Eq. (11) depends on the estimate of the underlying CNT, which often can be off by many orders of magnitude [17,18]. In our case however, we are only interested in the deviations arising from the pressure-effect. By renormalizing Eq. (11) we get a reasonable and mostly model-independent estimate of this deviation [12]: Fig. 1 shows the change of the rate as a function of the ratio of carrier-gas to vapor molecules, N c /N, which for perfect gases is the same as the ratio of carrier-gas over vapor pressure, p c /p. The calculation was performed for argon at 50 K and a supersaturation of S = 869 [19] and the latent heat h was calculated via the Clausius-Clapeyron relation. Fig. 1 also shows the individual contributions coming from nonisothermal corrections, which always have a positive effect on the rate, and pV -work, whose effect is always negative. The competition of both terms first leads to an increase in the rate with increasing carrier-gas pressure due to better thermalization. Then, the penalty of pV-work takes over and leads to an overall decrease of the rate.
We have performed standard molecular dynamics (MD) simulations of Lennard-Jones (LJ) argon nucleation at 50 K (N Ar = 343, S = 869) and 80.7 K (N Ar = 512, S = 10.5) with LJ-helium as carrier-gas in order to verify the predictions of our model. Details of the simulation are similar to Ref. [16]. The rates were analyzed using a method based on mean firstpassage times [20]. The studied N c /N-ratios range from 1 to 20. We have renormalized the simulation data to the first point of each series because we are only interested in the deviation arising from an increase of the amount of carrier-gas. Fig. (2) shows the simulation result together with the prediction of Eq. (12), which is also renormalized by the same fixed value. The agreement is quite remarkable. At 50 K, we only observed a slight increase in the rate with increasing amount of carrier-gas. On the other hand, the rate already starts to drop at a much smaller ratio of N c /N = 4 at 80.7 K, again following the theoretical prediction remarkably close.
We now take a closer look at the influence of temperature on the pressure-effect. Fig. (3) again shows the pressure-effect for argon, this time for three different nucleation temperatures. We took the supersaturations at each T to correspond to approx. the same rate from an earlier work [19]. The nonisothermal effects, Eq. (10), are only weakly depending on the supersaturation of the system. The pV -work term, however, gets more and more pronounced the higher the temperature. The reason for this is the huge change in the equilibrium vapor pressure by almost three orders of magnitude with increasing T . Hence, the same p c /p-ratio corresponds to a much higher total pressure in the system and the pV -work contribution is taking over earlier than at lower T . We have also framed a region of ratios that can be encountered experimentally [17]. In a typical experiment [4,5] to study the pressure-effect, one would fix T and S and vary the total pressure by increasing the amount of carrier-gas. We can do the same in Fig. (3), going from 3:1 to 20:1. At 50 K we observe a mild increase of the rate of about a factor of 2. There is practically no change in the rate at 65 K, at least none that would be detectable by available experimental techniques. Finally, at 80.7 K, we only observe a strong decrease of the rate of up to 2 orders of magnitude. Thus, we understand clearly how it is possible to observe only a positive, a negative, or no effect at all for the same substance. Both the sign at any given p c /p-ratio and the magnitude of the effect will strongly depend on the equilibrium vapor pressure (thus on the substance and temperature) and the experimental window of accessible p c /p-ratios.
Finally, we discuss some of the generic conclusions that can be drawn from our model with respect to different experimental situations. Since experiments greatly vary in the preparation, parameters, rate window, evaluation method etc., a careful evaluation requires a comprehensive analysis that we leave for a future work. In general, we can distinguish between two different limiting behaviors of the pressure-effect, depending on the relative influence of the pV -work term and, therefore, the total pressure p tot = p + p c . If p tot v l k B T ≪ ln S, the contribution of pV -work can be neglected and the only noticeable contribution comes from nonisothermal effects. In that case, the possible influence of the pressure-effect follows the dashed line in Fig. 1 and the only effect we could observe in an experiment would be a slight increase in the rate with increasing amount of carrier-gas. Note also that we would not observe any pressure-effect at all if we already started from high molar fractions of carrier-gas. Interestingly, these are the conditions that we typically find in nucleation pulse chambers [21], where the total pressure varies only very little around 1 bar and where carrier-gas fractions are high. Our analysis thus justifies why no effect is found in these experiments. In the opposite limit, when p c /p is large and p tot v l k B T ∼ ln S, the system is perfectly thermalized but the dominating pV -work leads to a significant overall decrease of the rate. Moreover, we can make an estimate of the changes in the critical size in terms of the CNT estimate [11], again replacing the chemical potential by the effective one, Eq. (6). Eq. (13) shows that a further increase of the carriergas pressure (or similarily p tot ) would also lead to an increase in n * . In an experiment, this might be noticeable by an increase in the slope of a nucleation rate isotherm, which is connected to the critical size via the nucleation theorem [14]. However, care must be taken in that kind of analysis because the nucleation theorem in principle requires that the carriergas pressure p c is also fixed.
In summary, we have presented a simple model that is able to shed light on one of the biggest puzzles in current research of vapor-liquid nucleation: the "pressure-effect". The investigation of this effect is often entangled between experimental uncertainties, wildly different experimental conditions and procedures, and theoretical interventions on different stages of the modeling of nucleation and growth. We have presented a physically very appealing way to disentagle most of these ambiguities by properly incorporating the presence of a carriergas into CNT. Simulation results corroborate the validity of the model quite impressively. Still, we cannot discard the influence of other factors on the observed pressure-effect. Nevertheless, these factors, if applicable, can be added easily to our model as secondary contributions to the more fundamental and inevitable physical roles that the presence of an ambient carrier-gas plays in nucleation and which are accounted for in our model. For example, it is straightforward to include nonidealities of the vapor and carrier-gas as well as the compressibility of the liquid, which surely will play a more prominent role the higher the pressure. Another possible influence we have neglected here is a change in the kinetic prefactor. Finally, we deliberately separated the influence of a truly inert carrier-gas pressure from other effects such as binary nucleation or surface adsorption. It is somehow misleading to include these under the same "pressure-effect" tag because even though the strength of them may depend on pressure, their origin certainly is not the pressure of the carrier-gas. In any case, the insights provided by the model presented in this letter will undoubtedly be very helpful to quantify and remove the influence of the carrier-gas on experiments. This opens the door to a more accurate evaluation of nucleation rates, which has important implications on many atmospheric and technological processes.