We present a model of insoluble particle activation based on a modification of the Köhler equation in which we introduce a term based on the activity of water adsorbed on the particle surface. We illustrate the model by application to activation data from black carbon (BC) particles. We parameterize the model using a free energy of adsorption that reflects the relative affinity for water vapor adsorbing on either the BC surface or the adsorbed water layer. This enables the parameterization of either chemically modified (hydrophilic) or graphitic (hydrophobic) BC. Several features of a suite of carbon activation data are captured by the model. In particular, upper and lower bounding curves are predicted for activation supersaturation as a function of diameter. We show a large body of recent activation data that all fall within these bounds. The model also predicts that activation of BC aerosol leads to activation diameters from 3 to 10 times smaller than activation of soluble particles of identical dry diameter. The activation of smaller particles may be expected to impact the size distribution of resulting cloud droplets and thus the aerosol first indirect effect on climate. Finally, we compare activation of BC aerosol as calculated with this model to activation of mixed particles of BC and ammonium sulfate. It is shown that for some range of the adsorption free energy a hydrophilic BC aerosol is predicted to activate at lower supersaturation than comparable mixed aerosol of low mass fraction in the soluble component, indicating the utility of a model of mixed particle activation based on the adsorption of water to form an interfacial solution.
 The impact of aerosols on the radiative properties of the atmosphere has been the topic of study for many years [Penner, 2001; Twomey, 1974]. There has been considerable recent activity on this question [Lohmann and Feichter, 2005], as well as the commensurate impact of aerosols on the hydrological cycle [Ramanathan et al., 2001]. The mechanism and magnitude of this impact is determined primarily by the interaction of the aerosol with water vapor in the atmosphere. The optical properties and number and size distribution of an aerosol population vary dramatically depending upon physical state, e.g., dry (soluble or mixed soluble and insoluble particles below the deliquescence point, insoluble hydrophobic particles exhibiting little water adsorption) wet but unactivated (soluble or mixed particles in a deliquescent state, insoluble hydrophilic particles with appreciable adsorption of water) and activated CCN (any particles which, through the uptake of water, have achieved a sufficient radius to grow without bound given an unlimited reservoir of water vapor).
 In this paper we focus on the activation of small insoluble particles to form CCN. The classic theory employed for the activation of soluble particles is Köhler  theory [Chýlek and Wong, 1998]. In the original formulation the equilibrium vapor pressure over a liquid droplet as a function of diameter is calculated as a product of two components of the liquid activity. The first component is derived from the Kelvin equation and accounts for the increased vapor pressure over a solution composed of droplets whose diameter ranges up to 100–200 nm. The formula is parameterized in the liquid surface tension and molar volume, and in the form for pure water is equivalent to early expressions deriving the activation diameter for water in homogeneous nucleation [Pruppacher and Klett, 1978]. The second component accounts for the lowered vapor pressure due to the activity of an aqueous solution formed from soluble components of the aerosol. This component is parameterized in the solution concentration, which yields a formula for the particle diameter as a function of water uptake. Taken as a product the formula produces the vapor pressure of the droplet as a function of diameter and exhibits a maximum at the activation supersaturation and diameter.
 Modifications of the classic formula for application to more complex aerosol have been presented. In one set of examples the formulation of the solution activity is modified to account for the activation of a particle containing many soluble components [Seinfeld and Pandis, 1998]. In another the concentration dependence on diameter is modified to account for an insoluble core in the activation of mixed particle single [Pruppacher and Klett, 1978], or multicomponent solutions [Broekhuizen et al., 2004]. Recently, Kreidenweis et al.  presented a modification with which the effects of high vapor pressure deliquescence of relatively insoluble species is incorporated in the calculation of CCN activation and Laaksonen et al.  have explored a number of other potential mechanisms for the modification of the liquid activity. Taken as a whole these studies and many others have demonstrated the ability to model and calculate the activation of soluble or mixed aerosol with a wide range of chemical composition and solution complexity.
 A number of models of insoluble particle activation also exist, broadly distinguished as two types. In one model [Fletcher, 1958] the free energy of formation of a nucleating water drop is modified by the interaction with an insoluble surface. The interaction is expressed as a function of the contact angle of water with the surface, which is in turn a function of the relative surface tension describing interfaces between two condensed phases and one gas phase. A considerable literature addresses the various assumptions implicit in this model. It has been successfully employed to describe activation of carbon particles at very high supersaturations [Kotzick et al., 1997]. Significant modifications to the original Köhler model have also been applied to insoluble activation, including a formulation wherein the original formula of Fletcher are cast as a liquid activity in a Köhler formula [Dusek et al., 2006]. In another modification an additional term in the activity is proposed to account for surface charge effects as a means to lower activity and enhance activation of a solid particle [Wexler and Ge, 1998]. Finally, changes in the applicable surface tension due to organic surfactants on the insoluble particle surface are incorporated into the Kelvin component of the activity [Abdul-Razzak and Ghan, 2004]. These models of insoluble activation have been successfully used to model data. They differ primarily in the physical mechanism assumed to govern the interaction of water with the particle surface. The number of different mechanisms investigated in the papers discussed above indicates the uncertainty that remains regarding the proper mechanism or mechanisms to use. This question is central to determining which physical parameters are required to properly model insoluble activation, acquiring measured values of these parameters, and the extrapolation of any given model over the space of parameters required by any sophisticated application to activation of real particles in the atmosphere.
 In this paper we present another such modification of the classic Köhler model applied to insoluble particle activation. We introduce an additional term based on the activity of water adsorbed on the particle surface. The role of adsorption in the uptake of water by aerosol has been considered before in the deliquescence of soluble aerosol [Romakkaniemi et al., 2001] and as the assumed mechanism by which the critical embryo is formed in classic heterogeneous nucleation. Here we consider the situation where water uptake by adsorption onto the surface is the primary mechanism of uptake and can lead to sufficient uptake to achieve activation.
 We use the Brunauer, Emmett and Teller (BET) model of multilayer physical adsorption [Adamson, 1990]. The model is parameterized in the free energy of adsorption, determined experimentally from the measurement of adsorption isotherms which describe the magnitude of water taken up by the surface as a function of the liquid activity. We then couple the multilayer adsorption isotherm with the Köhler model through the introduction of this new activity dependence. We can then determine the particle diameter and supersaturation at activation as a function of the insoluble particle dry diameter and adsorption free energy.
 We apply the model to the activation of black carbon (BC) aerosol. In this paper we use the term pure BC to denote a particle free of a soluble component, but not necessarily graphitic, as the surface may be modified to a state of variable hydophobicity/hyrdrophilicity by chemical reaction. We calculate activation using the model presented here and compare the results with measurements of activation supersaturation and diameter observed in a number of laboratories. Several features of the suite of activation data are captured by the model. In particular, given an assumption of uniform coverage by the adsorbed layer, upper and lower bounding curves are predicted for activation supersaturation as a function of dry diameter. We show a large suite of recent activation data which all fall within these bounds. Finally, the model predicts that activation of BC aerosol leads to activation diameters from 3 to 10 times smaller than comparable soluble activation, which is of interest within the context of the size distribution of resulting clouds and the aerosol indirect effect on climate.
 In this first demonstration of the theory we use a very simple multilayer adsorption isotherm. Much more complex and realistic models for adsorption, which exist for many insoluble materials, are currently being explored. We also derive a diameter dependence on uptake where complete wetting of the particle is assumed, such that the insoluble core is coated with a water shell. Other morphologies of water uptake, such as an external model where a water drop grows in contact with the insoluble particle, limited in uptake by the adsorption isotherm, are also being explored.
 The paper is arranged as follows. We begin with a discussion of the formula and parameterization we use for the adsorption of water in section 2. We present the adsorption-based theory of activation in section 3. In section 4 we discuss model calculations and comparison with data. We discuss calculations for pure BC activation in section 5.1 and comparisons with mixed particle activation in section 5.2. We discuss conclusions of this work in section 6.
2. Adsorption of Water on Carbon Surfaces
 We use the BET multilayer equation of state for the adsorbed water layer [Rudzinski and Everett, 1992]. The equation is very common, and has been derived both thermodynamically [Hill, 1986], as a lattice problem introducing a configurational entropy, and kinetically [Adamson, 1990], as a steady state solution to successive Langmuir like adsorbed layers interacting through adsorption and desorption. The typical form of the equation is
where θ is the thickness of the adsorbed water, in layers, x is the water activity and c is a temperature-dependent parameter. The water activity is defined as the vapor pressure of water over the surface divided by the equilibrium vapor pressure over a flat water surface, x = P/Po(T). Equation (1) may be rearranged as a function of the water uptake as
 The constant c is a Boltzman term in the two adsorption energies of the system, the adsorption energy for a water molecule on the solid surface minus the energy of adsorption onto the water multilayer, c = RTln(ΔEsur − ΔEvap), where each energy difference is taken relative to the gas phase. A prefactor term (the ratio of the two surface lifetimes in the kinetic derivation) is usually assumed to be of order 1, resulting in an exponential temperature dependence of c with a temperature-independent constant. We assume instead that c is related to the free energies of the system
where ΔG is the difference in energies as defined above, (ΔGsur − ΔGvap). The vaporization free energy of water, ΔGvap, can be assumed to govern the interaction of water with the adsorbed water multilayer in this problem such that if c can be determined experimentally, the surface adsorption free energy, ΔGsur, is defined here and can be calculated. If the temperature dependence of c can be determined from experiment, then an entropy and enthalpy may be separately calculated. The energy term is well defined as the difference in the enthalpy of vaporization and adsorption. The entropy term is the difference of vaporization and adsorption entropies, but is also a function of the configurational entropy, or the statistics of occupation of the surface, which has no bulk counterpart. In this paper we calculate and refer to the free energy difference, ΔG, as shown in equation (3). Although this is not the usual form of the c constant we have had considerable success with this approximation in modeling the properties of the so-called quasiliquid phase in a number of systems [Henson and Robinson, 2004] with ice a particular example [Voss et al., 2005].
 We show calculations of the uptake of water on a solid surface at 293 K as a function of the adsorption free energy in Figure 1. Equation (1) is used, where the uptake is plotted as a function of the liquid activity. Water uptake in units of areal density is calculated from the number of layers, θ, multiplied by the areal density of water, 6.04 molecules/nm2. The adsorption free energy is calculated from equation (3) and therefore given relative to adsorption on the water layer, i.e., negative free energy indicates a surface interaction more hydrophilic than that with water, positive indicates an interaction more hydrophobic and zero a surface interaction identical to that with the liquid water layer. The difficulties in using any given equation of state are well known [Rudzinski and Everett, 1992] and can arise from surface ensembles of interaction sites of different strength, complex surface topography or a host of other complications. What we wish to accomplish here is to look at a representative set of free energies of adsorption characterizing the interactions over an applicable range of uptake. These calculations span 4 orders of magnitude in uptake and range from strongly hydrophobic to more hydrophilic than a pure water surface. The range of free energies used in Figure 1 is from −5.2 to 15.7 kJ/mol.
 One finds a similar range in the uptake of water by carbon surfaces in the literature. In three specific examples measured uptake ranges from 10−2 water molecules/nm2 on graphite reduced under hydrogen at high temperature [Miura and Morimoto, 1991] and 100 water molecules/nm2 on clean graphite [Miura and Morimoto, 1994] to 101 water molecules/nm2 on deposited kerosene soot [Popovicheva et al., 2003], all at the same 50% relative humidity and similar temperature.
3. Adsorption Theory of Activation
 The activation model is derived from the Köhler  model for the activation of soluble solid particles [Chýlek and Wong, 1998], where a growing particle is initially limited in the water that can be taken up by the elevated equilibrium vapor pressure over the curved surface
where P(D) is the equilibrium water vapor pressure over the particle as a function of the particle diameter, D, Po(T) is the equilibrium vapor pressure over a flat water surface, and xKEL is the activity as a function of D, with xKEL > 1. The activity as a function of diameter is derived from the Kelvin effect and is a function of the properties of water as [Chýlek and Wong, 1998]
where M is the molecular weight, σ the liquid/vapor surface tension, and ρ the density of water. The temperature is given by T, R is the gas constant and the exponential can be calculated as exp(0.66/(DT)) with D in micrometers and T in K [Seinfeld and Pandis, 1998].
 In classic Köhler theory the growing particle is a solution of solute in water, and the solute lowers the solution vapor pressure. The resulting equilibrium vapor pressure is then a function of both the solution activity and the effect of the curved surface. In this application we substitute the liquid activity of the adsorbed phase. Again in analogy with Köhler theory the resulting vapor pressure dependence of the system is expressed as
Rearranging equation (6), taking the logarithm and substituting the equation for xKEL, leads to
This is the typical final form of the Köhler equation, with xBET defined by equation (2) and substituted for the solution activity.
 In this first activation model we assume that the surface is completely wetted by the adsorbed layer of water. This is a first approximation and sure to fail for very hydrophobic adsorption free energies. We are currently working on activation models which do not make this assumption, and will discuss this further in section 4 in the context of some very high activation supersaturations observed for BC aerosol. Given the assumption of complete wetting we then use the thickness of the adsorbed water layer, equation (1), in the activation theory in the same way the total solution volume determines a particle diameter in Köhler theory. We express the particle diameter, D, as the sum of the dry particle diameter, d, and twice the water uptake in layers multiplied by the diameter of a water molecule, D = d + 2θdw. Rearranging yields the thickness of the adsorbed layer of water as a function of the particle diameter
Substituting equation (8) into equation (2) gives the reduced water activity over the surface as a function of the growing particle diameter. Further substituting that into equation (7) yields the final activation formula as
Equation (9) exhibits a unique maximum in the activity as a function of the growing particle diameter, D, for each value of c. In analogy with Köhler theory this is taken to denote the activation supersaturation and diameter. This is shown by calculating the activation of a d = 50 nm BC particle at 293 K for three different values of the adsorption free energy ΔG (using equation (3)) in Figure 2. The values of ΔG were chosen to span the values illustrated in the calculation of Figure 1. The water diameter used was 4.59 Å. For comparison a calculation of the activation of a 50 nm ammonium sulfate particle is also shown in blue in Figure 2 [Seinfeld and Pandis, 1998]. The solid lines are the calculated percent supersaturation, S = 100(P(D)/Po(T) − 1). The negative dashed lines are the contributions to equation (9), in the same units, from the adsorption (black, BC particles) and solution (blue, ammonium sulfate particle) activity. The positive dashed line (blue) is the pure water activity calculated from the Kelvin equation, equation (5) [Seinfeld and Pandis, 1998]. The model enables a direct comparison of the supersaturation required to activate a BC particle characterized by a particular carbon-water interaction. A general result of the calculations is the increasing supersaturation, and decreasing activation diameter predicted for BC carbon as a function of the adsorption free energy and the very hydrophilic surface interaction required to approach supersaturation levels comparable to a soluble particle. Also of particular interest is the result that even for very hydrophilic interactions, such that supersaturation levels are comparable to that for a soluble particle, the activation diameter for the BC particle is still considerably smaller.
 The activation supersaturation as a function of the initial particle dry diameter may also be calculated, generating a supersaturation curve for a given interaction free energy. This is shown in Figure 3, again calculated at 293 K. The model predicts a limiting behavior in these curves, such that no matter how negative the interaction free energy (hydrophilic the interaction) values more negative than approximately −5.2 kJ/mol all fall on the lower bounding blue curve of Figure 3. Conversely, no matter how positive the interaction free energy (hydrophobic, although still wettable, the interaction) values more positive than approximately 33.6 kJ/mol all fall on the upper bounding blue curve of Figure 3. Although these bounding curves exhibit the same slope of supersaturation as a function of dry diameter, a single curve representing an intermediate interaction free energy does not exhibit this slope. This is shown by the five curves calculated with the intermediate free energies, ΔG = 0.0, ΔG = 5.6, ΔG = 11.2, ΔG = 16.8 and ΔG = 22.4 kJ/mol, labeled a–e, respectively. The activation supersaturation as a function of the initial particle dry diameter for soluble ammonium sulfate particles is plotted in green for comparison.
 The dependence of activation diameter on dry diameter as a function of the interaction free energy is shown in Figure 4. Again, the dependence is bounded by the blue curves, representing the same interaction free energies denoting both hydrophilic and hydrophobic saturation at −5.2 and 33.6 kJ/mol, respectively, and the black curves are calculations using the same intermediate energies, labeled a–e, as above. The activation diameter as a function of dry diameter is again also shown for soluble ammonium sulfate particles in green.
 The mechanism of the limiting behavior in the activation supersaturation and diameter are the same. The hydrophobic bound indicates a carbon surface–water interaction free energy which limits water uptake to the extent that the particle behaves as a bare BC particle until the supersaturation of a pure water drop of that diameter is reached, at which point activation is achieved, independent of the interaction energy. This can be shown taking the limit of equation (9) as c goes to zero. The hydrophobic bound in Figure 3 is identical to the homogeneous activation supersaturation as a function of diameter for a pure water drop, calculated using only equation (4), and follows the blue lower bound D = d in Figure 4. The hydrophilic bound indicates a sufficiently attractive carbon surface water interaction that the system initially behaves as a single layer Langmuir adsorption problem, monolayer coverage is achieved at low supersaturation and the activation behavior is governed by the uptake of water onto the water surface, again independent of the adsorption free energy. This can again be shown taking the limit of equation (9), this time as c goes to infinity.
 The activation diameter of BC particles activating at the same supersaturation percent as a soluble ammonium sulfate particle, calculated at 293 K, are shown in Figure 5. The activation diameter is plotted as a function of the adsorption free energy for the BC particles. The dashed lines are the activation diameter for three ammonium sulfate particles of 20, 50 and 100 nm dry diameter activating at a supersaturation percent of 1.72, 0.43 and 0.15, respectively. The solid curves are the calculated activation diameter for BC particles of increasing dry diameter, which activate at an equal supersaturation percent as the soluble particles represented by the intersecting dashed line. For example, the top curve is the activation diameter as a function of adsorption free energy for BC particles of dry diameter ranging between 67 and 350 nm, each activating at 0.15%, for comparison with the 355 nm activation diameter of a 100 nm dry ammonium sulfate particle. At each BC adsorption free energy, a larger dry diameter is required of the BC particle to achieve activation at the specified supersaturation%. Figure 5 enables an example comparison of activation diameters for an ensemble of BC and soluble particles activating at the same supersaturation percent, as would occur in the atmosphere. BC activation diameters are constant for low adsorption free energy, reflecting a similar behavior for a sub ensemble of relatively hydrophilic particles, and increase with varying nonlinearity above approximately 10 kJ/mol, until for the most hydrophobic particles activation (and dry) diameters are larger than the soluble particle. It is clear from Figure 5 that equivalent BC activation diameters are roughly half those for the soluble particles until this value of 10 kJ/mol. Considerable work remains to fully examine such parameterizations; however, Figure 5 serves to illustrate one goal of the work presented here. Through examination of detailed mechanisms useful definitions may arise; for example, an adsorption free energy of 10 kJ/mol might usefully serve as a measurable quantity describing the BC surface which defines hydrophilic and hydrophobic.
 The model presented here suffers from the usual questions and limitations concerning the use of bulk properties to model the thermodynamics of water at these small dimensions and high radius of curvature [Pruppacher and Klett, 1978]. In addition to the values of the surface tension and molar volume used in equation (5) we make the new assumption that the bulk liquid activity of water on these particles is applicable. We have recently discussed this point with respect to the quasiliquid layer of water on ice [Henson et al., 2005]. Experimental results both verifying and contradicting the applicability of various bulk properties at small dimension are available, with no consensus yet as to a systematic means to address this problem. We note that work toward understanding these limitations is ongoing, with a particularly applicable set of new calculations concerning the thermodynamics of condensation in classical nucleation theory appearing recently [Merikanto et al., 2007].
4. Comparison With Activation Data
 In Figure 6 we plot a suite of activation data on carbon particles from the literature and compare these data with the bounds predicted by the calculations of Figure 3. The activation data are taken from recent studies and represent a number of different types of carbon surface. The data set includes diesel combustion and diffuse flame from fuel with and without sulfur (solid and open blue circles and open inverted blue triangles, respectively [Lammel and Novakov, 1995]), flame (blue diamonds) and gas (blue squares) soots with dry and aqueous preparations (solid and open symbols, respectively [Lammel and Novakov, 1995]), white gas combustion (solid yellow circles [Hagen et al., 1989]) and soot from carbon discharge in air and ozone (solid green diamonds and triangles, respectively [Kotzick et al., 1997]). We also include data on combustion soot from diesel fuel with and without sulfur obtained at subsaturated conditions (solid and open red circles, respectively [Weingartner et al., 1997]). With one minor and one major exception all these data fall within the bounds of activation supersaturation as a function of dry diameter defined by the model and denoted by the solid blue lines of Figure 6. The minor exception is a few points from gas combustion experiments which fall just high of the upper bound, although following a similar slope.
 The major exception is the spark discharge produced particles [Kotzick et al., 1997] which exhibit very high supersaturation. In that study it was shown that the classic heterogeneous nucleation approach could be used to model the data. At this time these data cannot be considered consistent with the predictions of the model presented here. However, one can relax the assumption of complete wetting used here, invalid at any rate for a sufficiently hydrophobic surface interaction, and derive a model whereby the water taken up by the particle is condensed into a growing drop, external to the particle. This type of model uses the same surface adsorption free energy parameterization we have described here but produces a particle with activation characteristics much more like that of the classic heterogeneous nucleation mechanisms, where a liquid drop essentially grows in contact with the particle, rather than incorporating it. We are currently developing this model and will report on the ramifications to high activation supersaturatiton in a forthcoming paper.
 The results of fitting a few of the individual data sets from Figure 6 are shown in Figures 7 and 8. In Figure 7 the results from the activation of particles produced by the combustion of white gas are shown, labeled as in Figure 6. The black line is a calculation of the activation supersaturation as a function of dry diameter using a value of 12.5 kJ/mol for the adsorption free energy. The blue lines again denote the predicted upper and lower bounds in both Figures 7 and 8. The calculation captures the observed activation behavior very well over an order of magnitude in the supersaturation. In Figure 8 two independent measurements of the activation of particles produced by the combustion of diesel fuel with a sulfur additive are shown, labeled as in Figure 6. Although there is some scatter in the data, it is clear that there is good agreement both between the two laboratories and with the calculations using an adsorption free energy of 16.8 kJ/mol.
5.1. Results for the Activation of Pure BC Aerosol
 Perhaps the most important first result of the model is the small activation diameter indicated by the adsorption mechanism. A reduction of activation diameter by factors of 3 to 10 under conditions of variable supersaturation, or factors of 2 at constant supersaturation, will yield an ensemble of CCN particles which is initially much smaller than that expected for an equivalent size distribution of soluble particles. From the example of Figure 2, direct comparison to an ammonium sulfate particle indicates that even for hydrophilic BC aerosol, activating at similar supersaturations, the expected activation diameter of the BC aerosol is less than 100 nm, as compared to 300 nm for the ammonium sulfate. In addition, Figure 5 illustrates the robust factor of 2 difference for activation of BC of variable size at constant supersaturation. This difference in diameter may significantly change the light scattering properties of the initial CCN ensemble. The ultimate perturbation to the scattering properties of the ensemble will depend on the number of activated particles and subsequent CCN growth, which are the defining characteristics of the first indirect mechanism of radiative forcing [Haywood and Boucher, 2000].
 The upper and lower bounds to activation supersaturation indicated by the model include a significant fraction of recent activation measurements on carbonaceous aerosol, lending confidence to the applicability of the model. In addition, individual data sets may be accurately parameterized using this model. However, these activation data may also be modeled assuming some soluble component. This will be discussed in section 5.2.
 We have presented activation calculations based upon assumptions of the adsorption free energy here. A suite of both adsorption and activation measurements on the same sample type would constrain both the adsorption free energy and activation parameters and would provide a stringent test to these ideas.
5.2. Comparison of Pure and Mixed Aerosol Using This Model
 CCN containing insoluble aerosol are often observed in the atmosphere in a mixed state, containing both the insoluble core and a soluble component. While mixed aerosol formation requires some mechanism to combine the components, which may still require pure particle activation, it is still pertinent to compare model predictions for the activation of mixed particles with the pure insoluble model presented here. We compare activation curves for pure BC and ammonium sulfate particles and intermediate mixed particles in Figure 9. The activation curve for a hydrophilic insoluble particle (ΔG = −5.2 kJ/mol, black line) is shown in comparison with a pure soluble particle (ammonium sulfate, blue) and several mixed particles with variable soluble mass fraction ɛm = 0.495, 0.250 and 0.100 (dashed blue lines, labeled in Figure 9). The curves are plotted as the supersaturation as a function of the wet particle diameter. The calculation of activation for the mixed particles is based on a Köhler model wherein the particle diameter is a function of the solution activity determined by the soluble mass fraction and an insoluble core diameter [Seinfeld and Pandis, 1998]. All particles are normalized to a 50 nm initial dry diameter, and so include an insoluble core of variable diameter.
 The activation of mixed particles as a function of mass fraction at constant initial dry diameter is as expected, with the larger mass fraction of soluble component enabling activation at the lowest supersaturation, and lower mass fraction leading to successively higher supersaturation. The two models are based on different physical mechanisms for calculating the liquid activity and so it is not expected that the pure soluble and pure insoluble activation curves should represent two limits in the activation supersaturation. It is clear from Figure 9 that they do not, with activation on the pure insoluble particle predicted to occur at significantly lower supersaturation than mixed particles of soluble mass fraction lower than approximately 0.5.
 It does seem, however, that the two pure particle states should represent limiting curves to mixed particle activation, with the pure insoluble particle representing the limit of infinite dilution at sufficiently small soluble mass fraction. This should be achievable by incorporating the activity of the solution formed at the interface upon the adsorption of water into the BET isotherm. Work is in progress to construct such a model.
 Finally, how do the predictions of the two models compare with respect to observed activation data. Is it possible to distinguish pure insoluble particle activation from that of a mixed particle using activation data. This is illustrated in Figure 10 and the answer seems to be no. In Figure 10 the activation supersaturation as a function of dry diameter is again shown with the suite of activation data on BC aerosol shown and labeled as in Figure 6. The solid black lines are the limits given by the mixed particle model of activation, bounded below by the activation of a pure soluble particle, again using ammonium sulfate as an example, and bounded above by the activation of a pure water or insoluble, hydrophobic particle. Only the lower bounds distinguish the two models, and the distinction is not strong in comparison with this collection of data. Furthermore, while mixed particle models have been successfully used to describe individual measurements, the insoluble model used here has also been shown to adequately describe individual measurements, as in Figures 7 and 8, and so there does not seem to be a means, using only activation data, to distinguish between pure and mixed particle activation.
 The model provides a physically meaningful mechanism for insoluble particle activation which incorporates the reduction in liquid activity of adsorbed water on the surface. The model predicts a reduced activation diameter compared to equivalent soluble particles and an upper and lower bound to activation supersaturation which includes a considerable number of recent measurements on the activation of carbonaceous aerosol. In particular, the prediction of relatively smaller activated particles is broadly consistent with observations of clouds formed in regions of high aerosol loading and may provide an accurate and simplified means of incorporating these effects in large-scale simulations of the atmosphere.
 Comparison with mixed particle activation indicates that an extension of this model to include adsorption of water to form dilute interfacial solutions would be a valuable addition to the suite of mixed particle activation models. Finally, the model presented here is suitable for application to any system of insoluble particulate, and work is in progress toward applications to other materials, e.g., mineral dust.
 During the course of the review of this paper it came to our attention that R. Sorjamaa and A. Laaksonen will soon publish a paper in which they outline an approach to activation similar to that discussed here.
 The author wishes to acknowledge helpful discussions with Laura Smilowitz concerning this work. This work was funded by the Laboratory Directed Research and Development Project entitled “Resolving the Aerosol-Climate-Water Puzzle (20050014DR)” administered by Los Alamos National Laboratory.