When a droplet falls through air, various processes can lead to the collision of aerosol particles with droplets. Theoretical formulations of these processes generally assume a flow around a spherical droplet capturing spherical particles. The collision efficiency (E) is defined as the fraction of particles in the cylindrical volume swept out by a falling droplet that collide with the droplet. A collision efficiency of unity is realized when all the particles residing in the swept out volume of a droplet collide with the droplet. When the particles follow the airstream around the droplet, E is smaller than 1. E can exceed unity when particles are scavenged by wake capture. The coalescence efficiency is defined as the fraction of particles that are retained by the droplet when they collide with them. The product of E and coalescence is called the collection efficiency. Normally, it is assumed that a collision leads to the scavenging of the particle by the droplet so that the collision efficiency and collection efficiency are the same. Different processes have to be considered that cause deviations of the particle's movement from the airstream path and lead to the collision of aerosols with droplets . For the smallest particles, Brownian diffusion is the most important collision process. Brownian diffusion describes the random motion of aerosol particles resulting from collisions with carrier gas molecules. Brownian motion strongly depends on the particle size and is most important for small aerosol particles. Large particles are most efficiently scavenged by inertial interception and impaction. Inertial impaction occurs when a particle is unable to follow the streamlines around a falling droplet and, because of its inertia, continues to move toward the drop and is eventually captured by it . Interception takes place when a particle follows the streamlines around a falling droplet sufficiently closely to collide with it. The region of low collision efficiency between the small and large aerosol particles is known as the Greenfield gap. This gap may at least be partly closed when electric and phoretic effects contribute to particle collisions. Thermophoresis describes a net transport of particles in the presence of a temperature gradient in the air. Air molecules at a higher temperature have a higher mean velocity and therefore impart more momentum to a particle than colder ones. The momentum on the warmer side of the particle is therefore larger and moves particles from higher to lower temperatures. Since evaporation cools the droplets and induces a temperature gradient in the surrounding air, particles are attracted by droplets for relative humidity (RH) < 100 % because of thermophoresis. Diffusiophoresis arises in the presence of a vapor concentration gradient. In the case of an evaporating droplet, there is a flux of water molecules away from the droplet, compensated for by a flux of carrier gas molecules (mainly N2, O2) in the opposite direction (Stefan flow) to keep the total pressure constant. Thus, thermophoresis and diffusiophoresis have opposite effects. Under typical atmospheric conditions thermophoresis dominates diffusiophoresis for aerosol particles < 1 µm . Finally, in the case of charged particles and droplets, electroscavenging has to be considered as an additional collision process. Usually, collision efficiencies of each of these processes are formulated separately and added together to yield the total collision efficiency Etot. and proposed formulations for the collision efficiencies by Brownian diffusion (EBr), interception (Eint), and impaction (Eimp). give formulations for thermophoresis (ETh), diffusiophoresis (EDf), and electrophoresis (EEl). use a flux model to calculate collision rate coefficients for electric and phoretic scavenging. In the following, we will outline the formulations proposed for the different collision processes.

In the approaches by , hereafter referred to as P05, and , hereafter referred to as S83, collision efficiencies of Brownian diffusion, interception and impaction (EBr,I) are provided. They are calculated separately and added together. EBr,I=EBr+Eint+Eimp

Since the droplets evaporate inside the collision chamber, thermo- and diffusiophoretic forces also contribute to the collision efficiency. Electroscavenging has to be taken into account because particles and droplets are charged. We consider the formulations of and , hereafter referred to as A06, where collision efficiencies are calculated separately and added together to obtain the total collision efficiency due to phoretic forces (Eph): Eph=ETh+EDf+EEl, where ETh, EDf and EEl are the collision efficiencies due to thermophoresis, diffusiophoresis and electrophoresis, respectively.

Our CoLlision Ice Nucleation CHamber (CLINCH) is similar to the one used by for contact freezing studies with some modifications to observe the frozen fraction of droplets at different times. It is a continuous flow diffusion chamber which consists of two parallel plates separated by 1 cm with side windows for the detector. Both chamber walls are held at the same temperature and are covered with ice, leading to an environment that is saturated with respect to ice and subsaturated with respect to water. Relative humidity in the chamber depends on the chamber temperature. A droplet generator from Bremen University is placed at the center top of the chamber. The droplet generator contains a piezo element which can produce 80±3 µm diameter droplets with a frequency of 100 droplets per second . With this setting the distance between two successive droplets is about 2 mm when the droplet acquires its terminal velocity of 0.186 ms-1. The droplets are generated with pure water (Milli-Q, 18.2 MΩ) at a temperature of 281 K. The relaxation time for a droplet to reach its terminal velocity is 0.2 s, and the temperature relaxation time is about 0.6 s when the chamber is kept at 235 K. Figure  shows the droplet surface temperature (panel a), the difference between droplet surface and chamber temperature (panel b) and the evolution of droplet diameter (panel c) at 255, 245 and 235 K. Aerosol particles enter the chamber at the top in an airflow from both sides and can interact with the liquid droplets while passing through the chamber. The flow through the chamber is laminar and should not show any turbulence. The terminal velocity of the AgI aerosol is too low to contribute to the flow velocity. The fall velocity of the aerosol particles is therefore taken as the flow velocity averaged over the whole cross section of the chamber, which equals 0.017 ms-1 for an airflow of 1 L min-1 through the chamber. In CLINCH, aerosol particles and cloud droplets can collide, leading to freezing of the cloud droplets via contact freezing. With the modified setup, it is possible to observe the frozen fraction of droplets at lengths of 40 and 80 cm. The residence times of droplets at these two lengths are 2 and 4 s, respectively. Since the droplets have to cool down to the chamber temperature after injection, the residence time at the desired temperature is shorter (see Fig. ). In the case of the lowest investigated temperature of 235 K, it is reduced to 1.4 and 3.4 s for chamber lengths of 40 and 80 cm. At the end of the chamber, a condensation particle counter (CPC, TSI 3772) is connected to measure the concentration of the aerosol particles. In order to discriminate between water droplets and ice crystals, an Ice Optical Detector (IODE) developed in-house was used. In order to avoid the simultaneous presence of two droplets in the laser beam, a new laser was installed (402 nm, Schaefter + Kirchhoff laser Makroliniengenerator13LTM), providing a rectangular instead of a circular laser beam. The fall velocity of the droplets is 0.210 ms-1, calculated as the sum of the terminal velocity of the droplets (0.186 ms-1) and the flow velocity at the center of the chamber (0.024 ms-1) using the formula by and neglecting the temperature gradient term. The Reynolds number of the airflow is calculated to be 12 and the droplets' Reynolds numbers are about 0.65, which ensures that the chamber flow is not turbulent. With such conditions inside the chamber, the relative humidity around the column of droplets will increase only slightly due to the evaporation of droplets. Our calculations show that this increase is < 1 % and is too small to trigger deposition nucleation on the aerosol particles. showed that deposition nucleation on silver iodide particles can take place only when the relative humidity with respect to ice is larger than 105 %.

Silver iodide particles were produced by mixing 0.1 M potassium iodide and 0.1 M silver nitrate solutions: 10 mL of the potassium iodide solution was diluted with 80 mL distilled water, and 10 mL of the silver nitrate solution were added. The volume of the AgI precipitate was then reduced to 40 mL by decantation, and 60 mL distilled water was added to the solution. From this solution aerosol particles were produced by atomizing. The particles were then dried to RH < 10 % at room temperature. These dried particles passed through a mixing volume to obtain a relatively constant concentration of particles. At the exit of the mixing volume, a cyclone with 1 µm cutoff size was used and the particles were then passed to the differential mobility analyzer (DMA). The DMA column voltage was set to negative so that only negatively charged particles entered the airstream. The size distribution of AgI particles is lognormal with a mode diameter of 80 nm. Size-selected 200 and 400 nm silver iodide particles are in the downslope of the size distribution of particles which were produced. Thus, the contribution of double or triple charged particles in the outflow of the DMA was < 10 %. These size-selected particles were passed to the chamber via a concentration control system in order to select a particular concentration of particles. The aerosol concentration was measured at the end of the chamber with a CPC.

The droplets obtain a variable number of charges when the stream of droplets is injected from the droplet generator and the charge can be measured with various methods. We determined the charge off-line of the experiment by passing the droplet stream through a capacitor consisting of two parallel plates which were connected to a DC voltage supply. The droplet generator was placed exactly at the top edge of the plates. These two plates were kept at 6 mm distance from each other and a DC voltage was applied. Due to the presence of the charge on the droplets, the droplets can be either deflected toward the positively or negatively charged plate. Multiple measurements were performed at different times in order to obtain the average charge on the particles. Figure  shows the individual measurements of the charge on the droplets. The charge on the droplets varied from 0.16 fC (1000 e) to 80 fC (50 000 e). It remained the same once the droplet generator was turned on but could switch to a different value when the droplet stream was turned off and turned on again. The mean charge on the droplets was about 65 fC (39 000 e ± 20 000 e).

The collision ice nucleation experiments were conducted at temperatures between 261 and 236 K. Initially the chamber was evacuated for 5 min and then cooled to 258 K. To cover the walls with a thin layer of ice, the chamber was filled with distilled water for 10 s and then flushed out. The chamber was again evacuated for 3 min and the detector was mounted. When the desired temperature of the chamber was reached, the droplet generator was turned on and droplets were observed in the detector. This blank experiment without aerosol particles was performed at each temperature in order to ensure that there is no droplet freezing without particles. After the blank experiments, the aerosol flow was turned on and the actual experiment was performed. After completing the experiment for one temperature, the temperature of the chamber was lowered in steps of 2 to 3 K until the homogeneous freezing temperature was reached.

Panel a of Fig.  shows the dependence of the collision efficiency due to thermophoresis for T=263, 248 and 233 K, keeping the other parameters the same as used for Fig. . As the temperature decreases the effect of thermophoresis also decreases because the evaporation rate of the droplet decreases and therefore also the temperature gradient decreases. On the other hand, the collision efficiency by thermophoresis is also influenced by the decreasing relative humidity with decreasing temperature from 90.6 % at 263 K to 78.2 % at 248 K and finally to 67.8 % at 233 K because the chamber is kept at ice saturation conditions. To separate the influence of temperature from that of relative humidity, panel b of Fig.  shows the dependence of thermophoresis on temperature with the environmental relative humidity with respect to ice at 90 % and panel c shows this dependence with the environmental relative humidity with respect to water at 90 %. In addition, curves at T=298 K are also shown as dash dotted lines in panel c for a better comparison with other studies. The temperature dependence of W78 is much stronger than the one of A06. Since for our experimental settings, diffusiophoresis results in a repulsive force rendering the collision efficiency negative or 0, we do not show the temperature dependence of diffusiophoresis here. The combined description of thermophoretic and diffusiophoretic forces indicate that for our experimental conditions of evaporating droplets in the presence of rather small aerosol particles, thermophoresis should exceed diffusiophoresis . However, disagreement still exists between experiments and model predictions concerning the prevalent forces as a function of particle radius .

Figure shows the droplet charge dependence of electroscavenging for the formulations of A06 and W78. Calculations are shown for droplet charges of 5000, 10 000, and 50 000 e and a particle charge of 1 e. All other parameters are the same as for Fig. . Charges on droplets and particles are of opposite sign. Both formulations show the same charge and particle size dependence with a strong increase in collision efficiencies with increasing droplet charge and decreasing particle size. If the charges of droplets and particles were of the same sign, particles would be repulsed from the droplets and the collision efficiency would effectively be 0, since the formulations of A06 and W78 both assume point charges located in the middle of the particles and droplets and do not take into account image charge effects. When aerosol particles come close enough to water droplets, image charges on the conducting droplets can lead to attraction even if the charges on the particle and on the droplet are of the same sign . When the radial component of the flow carries the particle towards the droplet as fast as the particle is repulsed, then the particle will pass through the distance of maximum repulsion and in most cases collide with the droplet, as the image forces increase very rapidly at close distance. show in their Fig. 5c and d that for particles with diameters > 500 nm and charges of 5–500 e electroscavenging by droplets with 84 µm diameter and a charge of 500 e, does not depend on whether the charges of droplets and particles are of the same or opposite sign. Particles with diameters < 200 nm are strongly repulsed from the droplets when their charge is of the same sign as the droplet charge and strongly attracted when the charges are of opposite sign. For particles with diameters of 200 nm, collision efficiencies are larger by a factor of 2 in the case of opposite sign than in the case of the same sign. Moreover, for our experimental situation, image charges will diminish the difference between electroscavenging between particles and droplets with like and opposite charges. However, the effect might be smaller because the AgI particles carry only one elementary charge, resulting in a smaller image force compared to the situation shown in , and the droplets are highly charged, increasing the radius of repulsion that has to be overcome until attractive image forces set in.

Figure shows the total collision efficiency for different combinations of the theoretical formulations for the same experimental conditions as in Fig. . All combinations of collision efficiencies are dominated by electrophoresis for particle diameters < 100 nm and by impaction for diameters > 2 µm. In this range Brownian diffusion, electrophoresis and thermophoresis contribute significantly to the total collision efficiency. For 200 nm particles, the total collision efficiency is lowest (0.01) for the combination P05 and A06 and highest (0.02) for the combination S83 and W78. In our approach total collision efficiencies are obtained by adding up collision efficiencies of the different processes with values ≥0. Negative collision efficiencies were not considered since they lack physical meaning. In trajectory calculations , the simultaneous action of the different forces on the particle can be investigated. These calculations show that, e.g., for small particles, the total collision efficiency can be lower than the one by Brownian diffusion alone when Brownian diffusion is diverted by the repulsion of particles carrying charges of the same sign as the droplet . Table  shows the dependence of the total collision efficiency on droplet and particle sizes used in the experiments for P05 and W78 (red line in Fig. ). Other formulations show similar variations in total collision efficiency. In addition, it lists the sensitivity of the total collision efficiency to the density of the particle and variation in atmospheric pressure. For both these parameters, the calculated collision efficiency sensitivity is found to be negligible. It should be noted that impaction collision efficiency in S83 is normalized by the density of the particle, but this effect can only be seen for particles larger than 2 µm. The listed particle densities are the ones of silver iodide (5600 kgm-3) and potassium nitrate (2100 kgm-3).

The experimentally derived collision efficiencies are almost 1 order of magnitude higher than the theoretical ones. It is unlikely that the experimentally derived ones are too high by this amount. The assumption that every collision leads to droplet freezing can only result in collision efficiencies that are too high. A conceivable process that would result in an overestimation of the collision efficiency could be that droplet freezing would influence the velocity of the droplets in such a way that frozen droplets collide with liquid ones. However, considering the sequence of frozen and liquid droplets, such a bias does not seem to exists. Brownian diffusion is the main collision mechanism for small particles in the absence of charges and one of the dominating contributions to the total collision efficiencies for the 200 nm diameter particles investigated in this study. The formulation of S83 predicts higher collision efficiencies than the one of P05 for Brownian diffusion and combinations with S83 for total collision efficiencies yield higher values. However, it is unlikely that uncertainties in the theoretical formulations of this process can account for the total discrepancy between experimentally derived and calculated collision efficiencies. Contributions of impaction and interception to total collision efficiencies are more than 1 order of magnitude lower than the one of Brownian diffusion, and therefore uncertainties in these formulations are not likely to fill the gap between experimentally derived and calculated collision efficiencies. The assumption that the charge on the droplet is 50 000 e and of opposite sign to the one on the particles results in the highest expected value for collision efficiencies due to electric forces. Accounting for image forces would only increase collision efficiencies in the case of the same charges on particles and droplets. The processes with the highest uncertainties are the ones arising from thermophoretic and diffusiophoretic forces. While diffusiophoresis leads to a repulsive force and does not contribute to the total collision efficiencies under our experimental conditions of evaporating droplets, thermophoresis is attractive and a dominating contribution. In the combined treatment of electrical, thermophoretic and diffusiophoretic forces, collection efficiencies can be lower than when efficiencies are treated separately and added up. In trajectory calculations Brownian motion, electrical and phoretic processes are treated together. Since the diffusiophoretic force of an evaporating droplet is repulsive, it can counteract attraction by thermophoresis and Brownian motion. A study on the simultaneous effect of phoretic processes performed by showed that thermophoresis dominates diffusiophoresis for evaporating droplets in quasi-steady state conditions for vapor diffusion and heat conduction. Similarly, the same charges on particles and droplets will divert particles away from the droplets at distances where mirror charges are too weak to lead to attraction and decrease the collision efficiency . For the calculation of collision efficiencies spherical particle shapes have been assumed. This assumption is valid for liquid or glassy particles but not for solid ones, which have a complex morphology with significant deviations from sphericity. The drag on a nonspherical particle depends on its orientation, which in turn is affected by shear in the flow field. In a theoretical study, estimated the effects of oblate and prolate particle rotation in shear flow and the shape dependency of the thermophoretic force of evaporating 30 µm radius droplets. The results indicate that the orientation effects of the shear flow will tend to decrease the thermophoretic force on the particle toward the drop surface in the size regime where phoresis dominates because the nonspherical particle aligns itself with the streamlines and the velocity component of the phorestic force is minimized. investigated the collection of uncharged prolate spheroidal aerosol particles by 30 µm radius collector droplets. They found that such particles can be captured on the downstream side of the collector in the absence of attractive forces in contrast to the case of spherical particles. In the case of prolate spheroidal aerosol particles collected by charged 30 µm radius droplets, the collision efficiencies for particles having large aspect ratios are significantly lower than those for spherical particles when the Coulomb force is dominant. These studies indicate that deviations from particle sphericity decrease collision efficiencies for the experimental conditions of our study and cannot account for the discrepancy between measured and calculated collision efficiencies. The largest uncertainties are associated with the theoretical description of phoretic processes at low temperatures. It might be necessary to reassess these to obtain expressions that are in better agreement with experiments.

The efficiency of contact freezing depends on the efficiency of the collision process and the ability of the particle to act as INP. The most important heterogeneous ice nuclei identified in the atmosphere so far are mineral dusts. Size distributions of mineral dusts depend on the age of the air mass because larger particles are removed by gravitational settling. Mineral dust particles cover a large size range from 0.1 to 100 µm . In the coupled aerosol–climate model ECHAM5-HAM, which was used to investigate heterogeneous contact and immersion freezing, the mineral dust aerosol is represented by two lognormal modes with mass-median radii of 0.37 and 1.75 µm . Small particles collide with droplets mainly due to Brownian diffusion; large ones do so due to impaction. The predicted number of collisions varies by up to a factor of 3 for Brownian diffusion and impaction depending on the mathematical formulation that one chooses. Most importantly, in the highly relevant size range for ice nucleation on mineral dusts from 0.5–2 µm, calculated collision efficiencies are strongly reduced when a critical Stokes number is included in the formulation for impaction. In updrafts within clouds, a slight supersaturation typically persists, directing particles away from droplets due to thermophoresis, which is only partly compensated for by attraction due to diffusiophoresis. In downdrafts or when dry air is entrained in clouds, droplet evaporation mostly occurs at the cloud top and close to the edges of cumuli, which is the region where first ice in clouds is indeed observed (). Under such conditions, thermophoresis leads to attraction and may contribute significantly to the collision efficiency in the size range 0.1–2 µm. The two formulations for thermophoresis are very different in terms of particle size and temperature dependence. This term has to be reassessed to improve estimates of contact nucleation in models. In addition electric forces act on the particles, which may significantly contribute to the overall collision efficiency. Evaporating cloud droplets and aerosol particles released from evaporated droplets from the same region of the cloud are supposed to have like charges . For particles of sizes that act as INP, such as mineral dusts, the predominant effect of their charge, irrespective of sign, is an increase in the collision rate due to the short-range electrical image charge attraction . Layer clouds such as stratocumulus and altostratus are weakly electrified, producing droplet charges in the consequent gradients of the electric field on the order of 100 e on 20 µm diameter cloud droplets . Thunderstorm clouds are strongly electrified , with cloud droplets bearing elementary charges in the range of 10 000–100 000 e. Taking the effect of image charges into account will therefore increase the collision rate of particles with droplets even more. In summary, the collision efficiency of mineral dust particles with cloud droplets is most probably underpredicted in state-of-the-art aerosol–climate models, leading to an underestimation of the relevance of contact nucleation, especially in evaporating clouds.

Impaction scavenging of aerosol particles can occur in-cloud and below-cloud. Below-cloud scavenging leads to the removal of aerosol particles from the atmosphere between the cloud base and the ground due to precipitation. In addition to impaction scavenging, in-cloud scavenging also includes contributions from nucleation scavenging . Aerosol scavenging is usually described by the scavenging coefficient (s-1) defined as the rate of aerosol removal . In field measurements, the scavenging coefficient is usually calculated from measurements of the change in aerosol size distribution with rainfall . For very small and very large particles, there is mainly an agreement with theoretical studies. However, theoretical parameterizations often underestimate observed scavenging coefficients by 1–2 orders of magnitude for particles in the 0.2–2 µm diameter range, where collection efficiencies are lowest. Theoretical models predict Brownian diffusion as the dominating scavenging process of particles with diameters < 0.2 µm and inertial impaction as the main scavenging process for diameters > 2 µm. For aerosol scavenging in the particle diameter range of 0.2–2 µm, contributions from electric and phoretic forces are thought to be important. In the case of thermal equilibrium between the droplet and the environment, water vapor evaporation or condensation are the only factors determining the temperature gradient; thermophoresis and diffusiophoresis are supposed to act in opposite directions. However, in rain events, falling raindrops can have a different temperature from that of the ambient air and diffusiophoretic and thermophoretic forces will reinforce each other . Many theoretical studies on scavenging do not take phoretic forces into account, but even those which do are not able to explain the discrepancies between field and observed scavenging coefficients in the Greenfield gap . Most model parameterizations treat the collision processes separately and either assume that they act in series or calculate the total collision efficiency as the sum of individual collision efficiencies . With this approach, the net effect of repulsive and attractive contributions of forces acting on particles cannot be taken into account correctly. Figures and show that the collision efficiency formulations of thermophoresis of W78 and A06 are vastly different. They are derived for large and small particles, respectively, but in most studies applied to the whole simulated particle size range. Moreover, the temperature dependency of the formulations by A06 and W78 is very different, indicating large uncertainties here, too. From this, it can be concluded that phoretic forces give important contributions to the scavenging of aerosol particles in the accumulation mode and are most probably also a main source of uncertainties in aerosol scavenging predictions.