week ending PRL 101, 125703 (2008) PHYSICAL REVIEW LETTERS 19 SEPTEMBER 2008 Unraveling the ‘‘Pressure Effect’’ in Nucleation Jan Wedekind,1,* Antti-Pekka Hyva¨rinen,2 David Brus,2,3 and David Reguera1 1Departament de Fı´sica Fonamental, Universitat de Barcelona, Martı´ i Franque`s 1, 08028 Barcelona, Spain 2Finnish Meteorological Institute, Erik Palme´nin aukio 1, P.O. Box 503, F1-00101 Helsinki, Finland 3Laboratory of Aerosol Chemistry and Physics, Institute of Chemical Process Fundamentals, Academy of Sciences of the Czech Republic, Rozvojova´ 135, CZ-16502 Prague 6, Czech Republic (Received 21 May 2008; revised manuscript received 11 July 2008; published 18 September 2008) 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. 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 molecular dynamics simulations and might have important implications since nucleation in experiments, technical applications, and nature practically always occurs in the presence of an ambient gas. DOI: 10.1103/PhysRevLett.101.125703 PACS numbers: 64.60.QÀ, 82.60.Nh A gas phase can be supersaturated considerably beyond However, continuous improvements in the experimental its equilibrium condensation point before liquid drops form setups confirmed rather than remedied this undesirable spontaneously. The pathway of the phase transition is and elusive effect. blocked because microscopic droplets are thermodynami- Following the experimental evidence, theoretical works cally less favorable than the bulk vapor. Therefore, the have tried to explain the pressure effect with varying transition can only be initiated by rare fluctuations exceed- success. These approaches include changes to classical ing a critical size, called the critical nucleus. The forma- nucleation theory (CNT) accounting for nonideal behavior tion of such a critical nucleus is the limiting step in the of the vapor and/or carrier gas [6], treating the carrier gas transition and its frequency of occurrence is called the as not fully inert or the problem as binary nucleation [7–9], nucleation rate. Nucleation is behind most phase transition or analyzing its influence on cluster stability and the im- and plays a crucial role in atmospheric processes such as pingement rate [10], and many more. There remain many the formation of aerosols or the condensation of water contradictions and differences both in the direction and in vapor into clouds [1,2]. An accurate experimental evalu- the magnitude of the effect. These are complicated by the ation of the nucleation of atmospherically relevant sub- discrepancies between experimental results themselves stances and its correct theoretical prediction are essential and in the comparison of experiment and theory. for a better understanding of climate change and are the Here we present a simple yet physically very appealing subject of intense investigations [3]. model that resolves many of the apparent contradictions of Nucleation is highly sensitive to small changes in the the pressure effect. We take a deliberate step back and state variables describing the system, most notably to incorporate the presence of a carrier gas into classical temperature. However, condensation is inevitably con- theory in a most natural manner that accounts for the two nected with the release of latent heat. In experiments this primal contributions of the carrier gas: the efficiency of latent heat is removed by the presence of a large back- thermalization and the additional work that a cluster has to ground of carrier gas. This carrier gas should be noncon- spend for growing in its presence. These contributions have densing and chemically inert and should have no influence opposite trends and we show how this may be responsible on the nucleation except serving as the desired heat bath. for the existence of apparently contradictory results. Nevertheless, many experimental results suggest that there Nucleation theory [11] usually aims at calculating iso- can be an influence on the nucleation rate that can span thermal nucleation rates, taking the constant value of the some orders of magnitude, depending on the pressure and temperature and the idea of a noninfluential carrier gas for type of the carrier gas (see Refs. [4,5] and references granted. In CNT, the work of formation of a droplet of size therein). Unfortunately, the overall picture is far from clear. n at constant pressure p and temperature T is [11] Comparisons of experimental findings cover all possibil- 2=3 ities: no effect, increase, or decrease of the nucleation rate ÁGðnÞ¼nÁ þ s1n ; (1) with carrier gas pressure. There were some concerns of whether the pressure effect (PE) is not just an experimental which is the combination of a volume term related to the artifact, which cannot be completely ruled out in all cases. difference in chemical potential Á between the vapor and 0031-9007=08=101(12)=125703(4) 125703-1 Ó 2008 The American Physical Society week ending PRL 101, 125703 (2008) PHYSICAL REVIEW LETTERS 19 SEPTEMBER 2008 the liquid and a surface term needed to build the liquid between the energy increase due to latent heat and the 2=3 interface with area A ¼ s1n and surface tension [s1 ¼ energy removal through elastic collisions with vapor and 2 1=3 ð36vl Þ is the surface area per monomer and vl is the carrier gas molecules. The parameter volume per molecule in the bulk liquid]. The vapor pres- k T @AðnÞ p q ¼ h À B À sure is assumed as ideal and the liquid cluster is consid- 2 @n (8) ered as an incompressible spherical drop with a sharp interface and bulk liquid properties. The free energy has quantifies the energy released when a monomer is added to à à a maximum at the critical size n and its height ÁG is a cluster, which is the latent heat h per molecule (corrected k T=2 16 v2 3 by a small amount B ) minus the energy spent on ÁGà ¼ l : increasing the surface area AðnÞ against the surface ten- 3 Á2 (2) sion. The mean squared energy fluctuation removed by The isothermal steady-state nucleation rate then is collisions with impinging molecules is sffiffiffiffiffiffi à JCNT ¼ K expðÁG =kBTÞ; (3) N m b2 ¼ 2k2 T2 1 þ c B N m (9) where K is a kinetic prefactor. In CNT, the difference in the c chemical potential is given by for an ideal monoatomic vapor and carrier gas [16](m is Á ¼ kBT lnS À vlðp À peqÞ: (4) the molecular mass, N the number of molecules of the condensable). Equation (9) indicates that a large number of S ¼ p=p p Here, eq is the supersaturation and eq is the carrier gas molecules Nc and a light (small molecular mass equilibrium vapor pressure. The second term in Eq. (4)is mc) carrier gas are most effective for a good and fast arising from the pressure-volume work the liquid drop has thermalization. Finally, the influence of nonisothermal to perform against the ambient vapor pressure [12]. It is effects on the steady-state nucleation rate is determined typically small and hence commonly neglected but we by the combination of q and b: keep it here for clarity. We now naturally account for the 2 presence of an ideal carrier gas by noting that the cluster b Jnonisoth ¼ Jisoth: (10) must also perform pV work against the ambient carrier gas b2 þ q2 pressure, Wc ¼ pcVðnÞ¼nvlpc [13]. The work of forma- tion including pV work now reads We now obtain the full pressure effect of the carrier gas on the nucleation rate by combining Eqs. (10) and (7), 2=3 ÁGpVðnÞ¼nÁeff þ s1n ; (5) taking the latter as the isothermal but pV-corrected rate: where we have cast the pV contributions into an ‘‘effective b2 J ¼ J : chemical potential’’ PE b2 þ q2 pV (11) Áeff ¼ kBT lnS À vlðp þ pc À peqÞ: (6) Obviously, the magnitude of the rate calculated by Physically, this expression for the chemical potential Eq. (11) depends on the estimate of the underlying CNT, difference between the vapor and liquid phases can easily which often can be off by many orders of magnitude be derived from the thermodynamics of a system formed [17,18]. In our case however, we are only interested in by a condensable vapor plus ideal carrier gas. More spe- the deviations arising from the pressure effect. By renorm- cifically, the second term on the right-hand side of Eq. (6) alizing Eq. (11) we get a reasonable and mostly model- arises from the integration of the Gibbs-Duhem equation independent estimate of this deviation [19]: for the liquid, which has been considered as incompress- J b2 J ible, and where it is important to note that the chemical PE ¼ pV : J b2 þ q2 J (12) potential of the liquid has to be evaluated at the total CNT CNT pressure of the system [13,14]. Figure 1 shows the change of the rate as a function of the Replacing Á by Áeff in Eq. (2), the barrier becomes N =N ÁGà ¼ð16=3Þðv2 3=Á2 Þ pV ratio of carrier gas to vapor molecules, c , which for pV l eff and the -corrected rate perfect gases is the same as the ratio of carrier gas over follows directly from Eq.