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COAGULATION OF CIGARETTE SMOKE PARTICLES
PII: S0021-8502(98)00071-8
J. Aerosol Sci. Vol. 30, No. 4, pp. 533—548, 1999 1999 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0021-8502/99 $ — see front matter
COAGULATION OF CIGARETTE SMOKE PARTICLES R. J. Robinson and C. P. Yu* Department of Mechanical and Aerospace Engineering, State University of New York at Buffalo, Amherst, NY 14260, U.S.A. (First received 17 March 1998; and in final form 4 June 1998) Abstract—Experimental measurements on the deposition of cigarette smoke particles (CSP) in the human airways have produced results that are inconsistent with typical deposition data based on particle size. Previous work relating to hygroscopic growth indicates that hygroscopicity alone can not account for this discrepancy. The present study investigates coagulation of CSP modeled as a polydisperse-charged aerosol as a possible explanation. The results of the model more accurately predict the experimental coagulation data for mainstream CSP than models that treat CSP as a monodisperse or polydisperse-uncharged aerosol. An aerosol with an initial charge distribution based on Boltzmann equilibrium yields slightly larger coagulation rates than the mainstream CSP polydisperse-charged model. The numerical results indicate that the size and charge distribution of sidestream CSP, with a concentration of 10 particles cm\, remain stable. In 2 s, the size distribution of mainstream CSP, with a concentration of 10 particles cm\, shifts to a larger size while becoming flatter and wider. The diameter of average mass increases from 0.29 to 0.5 km. Numerical results confirm experimental reports for mainstream CSP, which indicate that the total number of charged particles increases with time and, in the early stages of coagulation, the amount of charge per particle cannot be estimated based on the particle size. This study shows that polydispersecharged CSP, allowed to coagulate for 2 s in the mouth, will not produce size distributions that yield the observed deposition of CSP. However, additional coagulation will take place as the CSP travels through the respiratory tract, which will be investigated in future work. 1999 Elsevier Science Ltd. All rights reserved
I N T RO DU CT I O N
Experimental measurements on the deposition of cigarette smoke particles (CSP) in the human respiratory tract have produced results that are inconsistent with typical deposition data. Deposition studies on monodisperse particles, other than CSP, show that for a given breathing pattern, deposition is dependent on particle size (Morrow et al., 1966a; Morrow and Yu, 1985). Fresh CSP (age 0 s) range in size from 0.04 to 0.5 km with geometric mean diameters of 0.11 km for sidestream CSP and 0.17 km for mainstream CSP (Ueno and Peters, 1986). Therefore, the expected deposition efficiency for CSP is approximately 25%. The experimental data on mainstream CSP report average deposition efficiencies of 47% (Hinds et al., 1983) and 82—97% (Mitchell, 1962; Dalhamn et al., 1968), which is higher than expected. The experimental deposition efficiencies for sidestream CSP, 7—20% (Postendorfer and Schraub, 1972; Hiller et al., 1982; Phalen et al., 1994) and 17—41% (McAughey et al., 1994), are within the range that would be expected for submicron particles 0.1—0.5 km. However, localized deposition of sidestream CSP in the tracheal bronchial airways was found to be indicative of particles in the size range 6.5—7.1 km (Phalen et al., 1994). The discrepancies between expected and experimental deposition efficiencies for CSP may be due to cloud behavior, particle charge or particle growth due to coagulation or hygroscopicity. Previous work relating to hygroscopic growth shows that a mainstream CSP with initial diameter of 0.4 km will grow to 0.6 km, and a sidestream CSP with initial diameter of 0.2 km will grow to 0.26 km (Robinson and Yu, 1998), which suggests that hygroscopicity alone cannot account for the discrepancy between expected and measured deposition
* Author to whom correspondence should be addressed. 533
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efficiencies. The present study investigates the effect of coagulation on the size and charge distribution of inhaled CSP. Monodisperse coagulation due to Brownian motion alone (Smoluchowski, 1917) predicts negligible growth during the time that CSP are inside the respiratory tract. However, experimental measurements on the coagulation rate of mainstream CSP indicate that monodisperse Brownian coagulation theory is not a satisfactory method for describing coagulation of CSP. Experimental coagulation data could not be found for sidestream CSP. Listed in Table 1 are the reported experimental coagulation constants, K, for mainstream CSP, which represent the rate of increase in the average particle volume (or mass for constant density particles) per unit time. Keith and Derrick (1960), Leonard and Kiefer (1972) and Keith (1982) determined K from the plot of the inverse concentration as a function of time. Chen et al. (1990) determined K using an empirical equation for coagulation of polydisperse aerosols which was dependent on the count median diameter. A fairly large discrepancy exists between these studies which may be due to variations in the initial size distributions and measuring techniques. With the exception of Keith and Derrick (1960), the studies listed in Table 1 report a K value significantly larger than that for Brownian monodisperse coagulation, K "8k¹C/3g, where, k is the Boltzmann’s constant,
¹ the temperature, C the slip correction and g the viscosity of the medium (Friedlander, 1977). Keith and Derrick (1960) predict a smaller growth rate than Brownian monodisperse coagulation which seems to be in error since the slowest theoretical growth rate is described by monodisperse aerosols coagulating due to Brownian motion alone. Despite the discrepancies in the K values, the studies listed in Table 1 imply that mechanisms other than Brownian motion are present during the coagulation of CSP. Davies (1988) suggests that an erroneously large coagulation rate could be due to deposition of particles by diffusion on the walls of the aging tubes. Other explanations include the effects of particle charge and polydispersity which would increase the coagulation rate. CSP is a polydisperse aerosol with standard deviations reported for sidestream CSP ranging from 1.3 to 1.7 and for mainstream CSP ranging from 1.2 to 1.5 (Okada and Matsunuma, 1974; Chang et al., 1985; Ingebrethsen, 1986; Uneo and Peters, 1986). Friedlander and Hidy (1969) applied a similarity transformation to the size distribution function for mainstream CSP to predict coagulation rates. Size distribution curves were reported and compared to those measured by Keith and Derrick (1960) for ages 30—140 s. The study concluded that the agreement between experiment and theory was fair and that the experimental measurements showed a significantly higher proportion of large particles than predicted by similarity theory. The similarity solution developed by Friedlander and Hidy (1969) assumes a self-preserving size distribution, reached after and long period of time, and would therefore not apply to the present task of predicting coagulation of inhaled CSP which would remain in the lung for less than 10 s. Furthermore, the theory predicts Brownian coagulation only and does not account for other effects such as the charge distribution.
Table 1. Experimental coagulation constants for mainstream cigarette smoke particles Investigator
K* (cm s\;10\) Reported
K (cm s\;10\)
Monodisperse
Keith and Derrick (1960) Leonard and Kiefer (1972) Keith (1982)
4.8 20.0 23.8
Chen et al. (1990)
13.3
11.0 7.5 10.8 9.7 8.6 11.0 8.5
d R (km)
t (s)
0.23 0.76 0.23 0.29 0.42 0.22 0.45
30 1.7 0.05
(d ) (d )
(d ) (d )
(MMD) (CMD) (MMD)
2.5
* Coagulation constants have been corrected based on Smoluchowski theory which defines K to be twice the slope of the line 1/N vs t plot, where N is the number concentration and t is the time. Rd "geometric mean diameter, d "diameter of the average mass, d "count mean diameter, d "mass
mean diameter, CMD"count median diameter, MMD"mass median diameter.
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It is well known that CSP contains electrically charged particles and the charge distribution of mainstream CSP has been reported in several studies (Whytlaw-Gray and Patterson, 1932; Sano et al., 1953; DallaValle et al., 1954; Hinkle et al., 1954; Holmes et al., 1959; Norman and Keith, 1965), most predicting a symmetrical charge distribution with a charge per particle of 1 or 2, and some predicating as much as 10 or more, with the number of multiple charges increasing with time. These studies are summarized in Table 2. Charge data for sidestream CSP could not be found. However, it has been postulated that the charging mechanism for mainstream smoke is chemi-ionization, due to the physical conditions of the combustion zone (Norman and Keith, 1965). Therefore, it is likely that sidestream CSP undergoes a similar ionization process and achieves a similar charge distribution. Additional charging will occur for sidestream CSP due to atmospheric ion capture. Davies (1966) shows that, in the case of symmetrically charged aerosols, the increase of the coagulation constant due to attraction will be compensated by the decrease due to repulsion. However, Davies (1966) further predicts that this compensation is only present for weak bipolar aerosols and that strongly bipolar aerosols will experience an increase in the coagulation rate. No study has been found that attempts to quantify the effect of the charge distribution on the coagulation rate of CSP. To accurately predict the growth of CSP due to coagulation, the general coagulation equation (1) must be solved for the particular charge and size distributions of CSP. The coagulation equation used in this study was first presented by Smoluchowski (1917) for Brownian motion and modified by Fuchs (1964) and Davies (1966) for charged particles. The resulting integro-differential equation cannot be solved analytically without simplifying assumptions. Zebel (1958), assuming that particle charge was unipolar and proportional to the particle volume, which eliminates the summation symbols in equation (1), presented an approximate solution for an aerosol with a trapezoidal size distribution. No study has been found that presents a solution to this equation for CSP modeled as a polydisperse aerosol with a bipolar charge distribution. The purpose of the present study was to solve the general coagulation equation, using simple numerical techniques, for CSP with a known initial size and charge distribution. The solution predicts the change in the size and charge distributions that would occur in the period that CSP might remain in the mouth before being inhaled (0—2 s). These results are required to accurately determine the size distribution of inhaled CSP. In addition, the charge distribution is required to determine the enhanced deposition of CSP in Table 2. Experimental charge distribution of mainstream cigarette smoke particles Age (min) 1—1.5
2 30—45
Charge distribution (%) Number 0 55 51 41 31 45
of charges per particle 1 2 '2 44 1.2 0 39 10 0.1 48 6.5 4.1 60 9.2 0.3 47 6.9 1.2
Number of charges per particle 0 1—10 '10 '20 46 43 12 2—5 13 65 21 2—5
Size (km)
Concentration (part/cm)
Investigator Norman and Keith (1965)
0.20 0.23 0.29 0.35 Ave
1.6 E7
Holmes et al. (1959) NR
NR
NR
Number of charges per particle 0 1—2 26 74
0.1—0.25
NR
45
Number of changes per particle 0 1 2 3 4 5 6—10 20 22 16 12 8 6 16
0.1—0.6
—
Hinke et al. (1954) Dalla Valle et al. (1954) and Whytlaw-Gray et al. (1932) Sano et al. (1953)
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the human respiratory tract due to particle charge, which has not been determined by any previous work.
TH E OR Y
The three mechanisms that affect collision rate are Brownian or thermal motion, electrostatic forces, and kinematic diffusion. The latter mechanism includes the effects of turbulence, laminar shear or velocity profile and differential sedimentation. CSP is known to be a polydisperse, charged aerosol which, when inhaled, experiences turbulent flow in the upper airways and parabolic laminar flow in the lower airways. The present study is intended to predict the initial distribution of the inhaled CSP, after coagulation in the mouth, and therefore will not include kinematic diffusion. The change in the size distribution function due to coagulation alone was first described by Smoluchowski (1917) for Brownian motion and modified by Fuchs (1964) and Davies (1966) for charged particles. The resulting equation describing the change in concentration, nS, of particles with size d and charge z in discrete form is given by I I S
1 I\ " b(z , z , d , d )n n ! b(z , z , d , d )n n , (1) T S\T G I\G T G S\T I\G T S G I T G S I 2 ! S T G T G where b is the collision frequency function. The first term on the right-hand side of equation (1) represents the increase in concentration due to collisions between particles of size d and G charge z with particles of size d and charge z , where z #z "z , and d #d "d . The T H U T U S G H I second term on the right represents the rate of decrease in concentration due to collisions between particles of size d and charge z with all other particles. The term d represents the I S I number of unit size spheres present in the agglomerate. Scanning electron photmicrographs indicate that mainstream cigarette smoke particles form clusters upon coagulation (Chen et al., 1990). The effective diameter, d , of a cluster with d size units can be determined from C I I the fractal equation (Oh and Sorensen, 1997) dnS I dt
d I d "d , I 1.3
(2)
where d is the diameter of the unit size sphere. The general form of the collision frequency function, b, neglecting kinematic diffusion is K b" , f !
(3)
in which K is the coagulation coefficient for Brownian motion and f is the ! correction factor that describes the relative effect of Coulomb forces on charged particles. The formulation of each term is discussed below. K depends on the thermal diffusion coefficients of each colliding particle and is given by (Friedlander, 1977) K
2k¹ C C G# H [d #d ], " G H d 3g d G H
(4)
where k is Boltzmann’s constant, ¹ is the absolute temperature, g is the viscosity of the medium, and C and C are the slip correction factors for particles with diameters d and G H G d , respectively. Smaller particles have larger diffusion coefficients and therefore yield H larger values of K . Polydisperse aerosols will have a larger K than mono disperse aerosols because the size distribution maximizes the two optimum conditions for collision, i.e. the larger particles provide a large surface area for collision to occur while the smaller particles have a greater thermal motion.
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The correction factor, f , in equation (3) due to particle charge is given by (Fuchs, ! 1964) f
(eW!1) " , ! y
(5)
where y is the ratio of thermal potential energy to electrical potential energy between the colliding particles given by 1 e y" z z , (6) 2ne k¹(d #d ) T U G H in which e is the dielectric constant of a vacuum and e is the value of an elementary charge. For attractive forces, f (1, increasing the growth rate and for repulsive forces, ! f '1, decreasing the growth rate. ! Given an initial size and charge distribution, equation (1) can be solved using simple numerical techniques to determine the size and charge distribution as a function of time for CSP particles coagulating in the mouth before being inhaled. Initial size and charge distributions are available from a number of sources as discussed in the previous section. N U ME RI CA L RE SU LT S F OR MAI NS TRE AM CSP
Equation (1) is solved numerically to determine the size and charge distributions of mainstream CSP as a function of time for a given initial size and charge distribution. The solution is used to illustrate the distinction between coagulation of mainstream CSP modeled as a polydisperse-charged aerosol and that of the simplified models for a polydisperse-uncharged or monodisperse aerosol. The models are then compared with experimental measurements made by Keith (1982) of mainstream CSP having the same initial size distribution used as the numerical input. Using the results of the polydisperse-charged model of CSP, several plots are created to illustrate the change in the size and charge distributions from 0 to 2 s in order to approximate the size and charge distribution of the inhaled CSP. These plots are also used to illustrate the relationship between particle size and particle charge. Numerical input for mainstream CSP The size distribution and concentration for fresh (age 0 s) mainstream CSP is taken from Keith (1982). As seen in Table 1, Keith (1982) provides measurements of mainstream CSP at an age of only 0.05 s. Therefore, Keith (1982) has minimized the error due to coagulation that could have taken place prior to the size measurements in the other studies listed. The method used by Keith (1982) involved the instantaneous fixation of CSP as it issued from the mouth end of the cigarette. The fixation technique used the reaction of the liquid particles with methyl 2-cyanoacrylate vapor to form a solid aerosol. Because the solid particles do not coalesce when they collide, any further coagulation after fixation could be observed. The particle size distribution of the original smoke was determined from the remaining single particles and is shown in Fig. 1 for an unfiltered cigarette. Keith (1982) estimated that the diameter measurements could be overestimated by 0.06 km. This includes 0.04 km from the 0.05 s initial aging time and 0.02 km from the addition of the methyl cyanoacrylate reactant in the particle. The apparatus was designed to minimize error due to loss of particles by wall deposition, however an estimation of this loss was not given. A total concentration of 2.86;10 particles cm\ was found for an unfiltered cigarette. The concentration was determined by counting all particles and correcting for the presence of doublets. An estimated error for the concentration was not given. The discrete size ranges used in the numerical solution are shown in Fig. 1 for mainstream CSP. For computational purposes, each size range is converted using equation (2) to represent the number of spherical size units needed to make up a particle with the given diameter. The diameter of the size unit is taken to be 0.1 km.
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Fig. 1. Input size distribution for mainstream CSP.
The input charge distribution is based on the experimental data listed in Table 2. Complete charge data for fresh smoke could not be found in the literature. Holmes et al. (1959) made measurements after 2 minutes and did not indicate a concentration, therefore this data cannot be used to estimate an initial charge distribution. The studies by WhytlawGray and Patterson (1932), DallaValle et al. (1954) and Hinkle et al. (1954) were made soon after generation but the study did not report an exact age nor the relative percentages of particles with 1 and 2 charges. Norman and Keith (1965) listed the percent of particles having charges of 0, 1, 2 and '2 for different sizes, however, measurements were made at 1—1.5 min, although coagulation was probably minimal due to dilution of the aerosol to 10 particles cm\. Based on these studies, it was assumed that fresh mainstream CSP is made up of particles containing 0, 1 and 2 charges. Particles with charges greater than 2 were assumed not to exist in the initial distribution as indicated by the experimental data (DallaValle et al., 1954; Hinkle et al., 1954) and the conclusion made by Norman and Keith (1965) that multiple charges are developed by coagulation and not by the original burning process. Although Norman and Keith (1965) concluded that the total number of charged particles increases with diameter, their data indicate that the amount of charge per particle is not proportional to particle size. Therefore, each size range is assumed to initially contain a symmetrical charge distribution with 45% neutral particles, 47% with $1 charge and 8% with $2 charges. Comparison of charged-polydisperse model of CSP with other models The results of the numerical solution to equation (1), based on the initial size and charge distributions described above agree fairly well with the experimental results of Keith (1982). The decrease in concentration with time, calculated numerically for charged and uncharged CSP, is plotted in Fig. 2 along with the experimental results of Keith (1982) and estimations based on monodisperse coagulation with constant and varying coagulation coefficients. The results indicate that the polydisperse-charged model yields a more accurate prediction of the behavior of coagulating CSP than the simplified models of polydisperse-uncharged or monodisperse coagulation. The differences between the models are quantified in Fig. 3, a plot of inverse concentration as a function of time, from which the coagulation coefficient, K, can be determined. Regression analysis results in a second-order polynomial for both models and for the experimental data from Keith (1982), indicating that the coagulation
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Fig. 2. Concentration with time for different models compared to experimental data for mainstream CSP.
Fig. 3. Inverse concentration with time for different models compared to experimental data for mainstream CSP.
coefficient is actually decreasing with time. However, as can be seen from the plot, K remains nearly constant, and can be approximated based on a linear best fit to the data. The K values are listed in Fig. 3 and indicate that the polydisperse-uncharged and polydisperse-charged models predict growth, respectively, of 1.6 and 1.8 times the rate of monodisperse coagulation, while Keith (1982) reported a K value of 2.7 times the rate of monodisperse coagulation. Figure 4 shows a plot of the diameter of average mass, d , as a function of time. In 1.4 s,
the numerical models predict growth ratios (d/d ), for an initial diameter d "0.29 km, of 1.52 and 1.59, respectively, for polydisperse-uncharged and polydisperse-charged models. The numerical solution for the polydisperse-charged model falls in the range of the experimental growth ratio of 1.59—1.66 estimated by Keith (1982). The lower value includes
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R. J. Robinson and C. P. Yu
Fig. 4. Diameter of average mass, d , with time for different models compared to experimental data
for mainstream CSP.
the error due to the mass of methyl cyanoacrylate reactant in the particle, as mentioned previously. Experimental and numerical models predict ratios which are larger than the 1.45 and 1.41 values predicted by monodisperse coagulation. Coagulation rates were also calculated for an aerosol with the same initial size distribution as mainstream CSP, but with an initial charge distribution based on Boltzmann charge equilibrium. A Boltzmann charge distribution is wider with a larger number of higher charged particles (Hinds, 1978) than that present in CSP. The results are included in Figs 1, 2, and 3 and indicate that a Boltzmann charge distribution yields a slightly higher coagulation rate (1.7E-9 cm s\) than charged CSP (1.54E-9 cm s\) but less than experimental measurements of charged CSP (2.38E-9 cm s\) made by Keith (1982). The results presented in this study indicate that the numerical model representing CSP as a polydisperse-charged aerosol more accurately predicts the experimental data than the simplified models of polydisperse-uncharged or monodisperse coagulation. Although the growth ratio compares well with experiment, the numerical results predict a lower coagulation coefficient than that reported by Keith (1982). The discrepancy could be due to the measuring technique. Keith (1982) varied the coagulation time using different tube lengths between the mouth end of the cigarette and the injection collar, where the particle sizes were fixed. It is possible that loss of particles due to deposition on the tube wall caused an underestimation of the measured concentration and therefore an overestimation of coagulation coefficient. This would also cause error in the size distribution, but the result could be either an overestimation or an underestimation depending on wether the loss was due to mainly to sedimentation or diffusion. No estimation of this error was provided. Another reason for the discrepancy could be an underestimation of the charge distribution. The experimental charge data listed in Table 2 indicates that larger charges could be present in CSP. More accurate experimental data for fresh tobacco smoke are needed to further investigate this possibility. Differential sedimentation could account for additional coagulation and should be considered in future studies. Size and charge distributions of inhaled mainstream CSP As a result of coagulation, the inhaled size and charge distributions will vary depending on the amount of time the CSP are held in the mouth. The size distribution frequency, based on the polydisperse-charged model, is shown in Fig. 5 for ages 0—2 s. The peak size changes from 0.2 to 0.35 km. However, d increases from 0.29 to 0.5 km, which agrees well with
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541
Fig. 5. Size distribution frequency for mainstream CSP modeled as a polydisperse charged aerosol for ages 0—2 s.
Keith (1982) as discussed above (Fig. 4). The results show a significant increase in the proportion of larger particles with time which agrees with the experimental results of Keith and Derrick (1960) but differs from the similarity solution presented by Friedlander and Hidy (1969), which applies only for long coagulation times. The size distribution flattens with time as the number of particles greater than 0.4 km increase and the number less than 0.4 km decrease. The results of the numerical solution presented by Zebel (1958) for a different aerosol also display this behavior of flattening with time and pivoting around a given size. Experimental data on the change in the charge distribution of CSP for times less than one minute are not available for a precise comparison. However, general trends with time and size can be observed from the studies listed in Table 2. The charge distribution frequency for CSP is plotted in Fig. 6 for ages 0—2 s. In 2 s, the proportion of neutral particles decreases from 45 to 34%, the proportion of singly charged particles remains constant at 47%, the proportion of particles with $2 charges increases from 8 to 15.5% and the proportion with $3 to $6 charges increases from 0 to 3.4%. No particles are formed with charges greater than $6 in 2 s. These results agree with measurements reported in Holmes et al. (1959) which show that the percentage of neutral particles decreases with time. Charge distribution relative to particle size is illustrated in Fig. 7. As indicated, particle charge does not increase with increasing particle size, which agrees with measurements made by Norman and Keith (1965) but differs from the assumption made by Zebel (1958). In addition, the numerical results indicate that the percent of neutral particles decreases with increasing size until 0.44 km, which agrees with Norman and Keith (1965), but reaches a minimum at 0.55 km and then increases with increasing size. Therefore, both numerical results and experimental measurements for CSP indicate that the amount of charge carried by a particle is not proportional to particle size. During the initial stages of coagulation, aerosol particles are both acquiring and losing charge as they collide with particles of the same and opposite charge, respectively. A large diameter particle could leave a larger charge range and enter a smaller charge range due to charge neutralization. Therefore, a particle’s collision history at the time of measurement, not the particle’s size, determines the amount of charge carried by the particle. Eventually, Boltzmann equilibrium is reached, in which the relationship z"2.37(d) is valid, where z is the particle charge and d is particle diameter (Hinds, 1982). For CSP in the atmosphere, equilibrium is reached in approximately 1.5 h.
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R. J. Robinson and C. P. Yu
Fig. 6. Charge distribution frequency with time for polydisperse charged model of mainstream CSP.
Fig. 7. Average charge per particle with time from the polydisperse-charged model of mainstream CSP.
DallaValle et al. (1954) report that the equilibrium charge increases with diameter, although actual data for CSP was not given. The size distribution frequencies for each charge range give insight into how particle charges are formed as a result of coagulation. The size distributions for larger charges are skewed toward larger sizes, which indicates that multiple charges are formed by the process of coagulation as previously assumed in the literature (Norman and Keith, 1965). Charges not originally present (3—6) are formed immediately and their size distributions broaden and skew toward larger sizes with time. Figure 8a illustrates this trend for particles with six charges. The larger sizes could not be formed by charge neutralization because particles with charges greater than 6 were not formed in 2 s. Therefore, the larger sizes have been formed by collisions between particles of similar sign. Figure 8b shows the change in the size distribution with time for particles with three charges. The trend is the same as that for
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particles with six charges, however, the larger particles could have been formed by collisions between particles with the same or opposite charges. The peaks and valleys found in the size distribution for charge 3 were also found in measurements made by Holmes et al. (1959), Hinkle et al. (1954) and Sano et al. (1953) and could be due to the quantum nature of charge levels. The percent of singly charged particles remains unchanged (Fig. 6) and the size distribution, shown in Fig. 8c, remains narrow and shifts slowly to the right indicating that particles are entering the charge range due to charge neutralization and leaving the charge range due to collisions between charges of the same sign at approximately the same rate. A similar trend is indicated by the size distribution of neutral particles, shown in Fig. 8d, however the net effect is a loss of particles to higher charge ranges. The slow rate of formation of larger particles indicates that the neutral particles are made up primarily of the original particles
Fig. 8(a) and (b). Size distribution frequency with time from the polydisperse-charged model of mainstream CSP for charge: (a) $6, (b) $3.
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R. J. Robinson and C. P. Yu
Fig. 8(c) and (d). Size distribution frequency with time from the polydisperse-charged model of mainstream CSP for charge: (c) $1, (d) 0.
in the neutral charge range and not by collisions between particles with the same but opposite charges. Three-dimensional plots of concentration as a function of size and charge are shown in Figs 9a, b and c*, for 0, 1 and 2 s, respectively. This information can be used to accurately predict the deposition of charged CSP in the human respiratory tract after coagulation in the mouth. Enhanced deposition is expected due to the increase in the average size and the presence of charged particles. Since the total concentration of particles entering the lung is on the order of 10 particles cm\, additional coagulation is expected to take place after entering the trachea. However, the magnitude of growth must be determined in conjunction with the decrease in concentration due to deposition. Furthermore, kinematic effects
* Figures 9a—c show only the distribution of the positively charged particles. An equal number of negatively charged particles exist for each charge range.
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experienced in the lung must be added to the coagulation model. These factors along with hygroscopic growth, cloud behavior and the effect of the resulting charge distribution on the deposition of CSP in the human airways will be investigated in future work using a previously developed deposition model (Chan and Yu, 1982; Yu, 1985). N U MERI CA L RE SU LT S F OR SI D ES TRE AM CSP
The sidestream CSP input size distribution was determined from the geometric mean diameter and geometric standard deviation measured by Okada and Matsunuma (1974), assuming a lognormal distribution. Each discrete size range was converted to a number of 0.1 km diameter size units using equation (2). The concentration of sidestream CSP was
Fig. 9(a) and (b). Concentration of mainstream CSP, modeled as a polydisperse-charged aerosol, at age: (a) 0 s, (b) 1 s.
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R. J. Robinson and C. P. Yu
Fig. 9(c). Concentration of mainstream CSP, modeled as a polydisperse-charged aerosol, at age: (c) 2 s.
estimated to be on the order of 10 particles cm, corresponding to the maximum estimated concentration in indoor air which ranges from 20 to 3000 kg m\ (Georghlou et al., 1991). Since no charge data are available for sidestream CSP, the same charge distribution used for mainstream CSP was assumed to exist for sidestream CSP. Therefore, each size range is assumed to initially contain a symmetrical distribution with 45% neutral particles, 47% with $1 charge and 8% with $2 charges. The numerical results indicate a negligible change in the size and charge distributions of sidestream CSP aged for 2 s. The total concentration is reduced from 1.00;10 to 0.996;10 particles cm\, while the charge distribution frequency remains unchanged. The possibility of underestimating the initial strength of the charged particles must be considered. If the sidestream CSP remained in a room for 1.5 h, Boltzmann’s equilibrium would be reached at which time the charge distribution would be slightly higher. However, by this time the aerosol would be stabilized and coagulation would have ceased. Therefore, the charge distribution used in this analysis appears to be reasonable and the results indicate that the charge is not strong enough to increase the coagulation of sidestream CSP at typical concentration levels. Treating sidestream CSP as a polydisperse charged aerosol yields the same results found using monodisperse coagulation theory which predicts a stable size distribution for aerosols with concentrations on the order of 10 particles cm\ (Hinds, 1982). CON CL US I O N
The general coagulation equation (1) was solved for mainstream and sidestream CSP modeled as a polydisperse-charged aerosol. For mainstream CSP, the results more accurately represent the experimental data (Keith,1982) than models treating CSP as polydisperse-uncharged or monodisperse aerosols. Polydisperse-charged mainstream CSP was found to have a slightly smaller coagulation rate than an aerosol having an initial charge distribution representing Boltzmann equilibrium. Using the polydisperse-charged model for mainstream CSP, the size and charge distributions were determined for ages 0—2 s, representing the amount of time the CSP would
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remain in the mouth before being inhaled. In 2 s, the peak size changes from 0.2 to 0.25 km and the diameter of average mass increases from 0.29 to 0.5 km. The size distribution flattens with time and a significant number of larger particles are formed which agrees with the experimental data of Keith and Derrick (1960) but differs from the similarity solution presented by Friedlander and Hidy (1969). The change in the charge distribution of mainstream CSP predicted by the numerical model is representative of the general trends with time and particle size found in the literature. The proportion of neutral particles decreases with time, agreeing with experimental results of Holmes et al. (1959). Specifically, in 2 s, the proportion of neutral particles decreases from 45 to 34%, the proportion of singly charged particles remains constant at 47%, the proportion of particles with $2 charges increases from 8 to 15.5% and the proportion with $3 to $6 charges increases from 0 to 3.4%. No particles are formed with charges greater than $6 in 2 s. Both numerical results and experimental measurements (Norman and Keith, 1965) indicate that the number of charges carried by a particle is not proportional the particle size. Furthermore, the percent of neutral particles decreases with increasing size until 0.55 km, agreeing with Norman and Keith (1965), but then increases with increasing size. Therefore, in the early stages of coagulation, the amount of charge per particle cannot be estimated based on the particle size. The numerical results for sidestream CSP modeled as a polydisperse-charged aerosol indicate a negligible change in the size and charge distributions in 2 s. Similar results are found with monodisperse coagulation theory which predicts a stable size distribution for all aerosols with concentrations on the order of 10 particles cm\ (Hinds, 1982). The results of this study indicate that accounting for polydispersity and particle charge in the coagulation calculation does not result in sizes large enough to account for the differences in expected and measured deposition efficiencies of mainstream CSP, nor the localized deposition in the tracheal bronchial region of sidestream CSP. More accurate charge data is needed for fresh CSP to improve the accuracy of the results. Additional factors such as hygroscopic growth, kinematic coagulation and cloud effect should be considered in future studies. The charge and size distributions resulting from these numerical calculations will be used to investigate the enhanced deposition, due to particle charge, of CSP in the human airway using a previously developed deposition model (Chan and Yu, 1982; Yu, 1985). REF ER E NCE S Chan, T. L. and Yu, C. P. (1982) Charge effects on particle deposition in the human tracheobronchial tree. Ann. Occup. Hyg. 26, 65—75. Chang, P. T., Peters, L. K. and Ueno, Y. (1985) Particle size distribution of mainstream cigarette smoke undergoing dilution. Aerosol Sci. ¹echnol. 4, 191—207. Chen, B. T. Namenyi, J., Yeh, H. C., Mauderly, J. L. and Cuddihy, R. G. (1990) Physical characterization of cigarette smoke aerosol generated from a Walton smoke machine. Aerosol Sci. ¹echnol. 12, 364—375. Dalhamn, T., Edfors, M. and Rylander, R. (1968) Retention of cigarette smoke components in human lungs. Arch Environ. Health 17, 746—748. DallaValle, J. M., Orr, C. and Hinkle, B. L. (1954) The aggregation of aerosols. Br. J. Appl. Phys. 3, 198—206. Davies, C. N. (1966) Aerosol Science. pp. 42—45. Academic Press, New York. Davies, C. N. (1988) Cigarette smoke: generation and properties of the aerosol. J. Aerosol Sci. 19, 463—469. Friedlander, S. K. (1977) Smoke, Dust and Haze, pp. 175—184. Wiley, New York. Friedlander, S. K. and Hidy, G. M. (1969) New concepts in aerosol size spectrum theory. In Proc. of the 7th Int. Conf. on Condensation and Ice Nuclei (Edited by Podzimek, J.), pp. 21—25. Academea, Prague. Fuchs, N. A. (1964) ¹he Mechanics of Aerosols, pp. 305—309. Pergamon Press, Oxford. Georghiou, P. E., Blagden, P., Snow, D. A., Winsor, L. and Williams, D. T. (1991) Mutagenicity of indoor air containing environmental tobacco smoke: Evaluation of a portable pm-10 impactor sampler. Environ. Sci. ¹echnol. 25, 1496—1500. Hiller, F. C. (1984) Deposition of sidestream cigarette smoke in the human respiratory tract. Preventive Medicine 13, 602—607. Hiller, F. C., McCusker, K. T., Mazumder, M. K., Wilson, J. D. and Bone, R. C. (1982) Deposition of sidestream cigarette smoke in the human respiratory tract. Am. Rev. Respir. Dis. 125, 406—408. Hinds, W. C. (1982) Aerosol ¹echnology: Properties, Behavior, and Measurement of Airborne Particles, pp. 233—244. Wiley, New York. Hinds, W., First, M. W., Huber, G. L. and Shea, J. W. (1983) A method for measuring respiratory deposition of cigarette smoke during smoking. Am. Ind. Hyg. Assoc. J. 44, 113—118.
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Hinkle, B. L., Orr, C. Jr. and DallaValle, J. M. (1954) A new method for the measurement of aerosol electrification. J. Colloid Sci. 9, 70—80. Holmes, J. C., Hardcastle, J. E. and Mitchell, R. I. (1959) The determination of particle size and electric charge distribution in cigarette smoke. ¹obacco Science 3, 148—153. Ingebrethsen, B. J. (1986) Evolution of the particle size distribution of mainstream cigarette smoke during a puff. Aerosol Sci. ¹echnol. 5, 423—433. Keith, C. H. (1982) Particle size studies on tobacco smoke. Beitr. zur ¹abakforschung 11, 123—131. Keith, C. H. and Derrick, J. C. (1960) Measurement of the particle size distribution and concentration of cigarette smoke by conifuge. J. Colloid Science 15, 340—386. Leonard, R. E. and Kiefer, J. E. (1972) Coagulation rate of undiluted cigarette smoke. ¹obacco Science 16, 35—37. McAughey, J. J., Knight, D. A., Black, A. and Dickens, C. J. (1994) Environmental tobacco smoke retention in humans from measurements of exhaled smoke composition. Inhalation ¹oxicology 6, 615—631. Mitchell, R. I. (1962) Controlled measurement of smoke-particle retention in the respiratory tract. Am. Rev. Respr. Dis. 85, 526—533. Morrow, P. E. and Yu, C. P. (1985) Models of aerosol behavior in airways. In Aerosols in Medicine. Principles, Diagnosis and ¹herapy (Edited by Moren, F., Newhouse, M. T. and Dolovich, M. B.), pp. 149—191. Elsevier Science, New York. Norman, V. and Keith, C. H. (1965) Charged particles in cigarette smoke. ¹obacco Science 9, 75—79. Oh, C. and Sorensen, C. M. (1997) Light scattering study of fractal cluster aggregation near the free molecular regime. J. Aerosol Sci. 28, 937—957. Okada, T. and Matsunuma, K. (1974) Determination of particle size distribution and concentration of cigarette smoke by a light-scattering method. J. Colloid Interface Sci. 48, 461—469. Phalen, R. F., Oldham, M. J. and Mannix, R. C. (1994) Cigarette smoke deposition in the tracheobronchial tree: Evidence for colligative effects. Aerosol Sci. ¹echnol. 20, 215—226. Postendorfer, J. and Schraub, A. (1972) Concentration and mean particle size of the main and sidestream cigarette smoke. Staub-Reinhalt. ¸uft. 32, 33—36. Robinson, R. J. and Yu, C. P. (1998) Theoretical analysis of hygroscopic growth rate of mainstream and sidestream cigarette smoke particles in the human respiratory tract. J. Aerosol Sci. ¹echnol. 28, 21—32. Sano, K., Fujiya, Y. and Sakata, S. (1953) A colloid study of tobacco smoke: particle size distribution and electric charge. J. Jap. Chem. Soc. Pure Chem Sec. 74, 664—668. Smoluchowski, M. v. (1917) Versuch einer mathematischen theorie der koagulationskinetik kolloider losungen. Z. Physik. Chem. 92, 129—168. Ueno, Y. and Peters, L. K. (1986) Size and generation rate of sidestream cigarette smoke particles. Aerosol Sci. ¹echnol. 5, 469—476. Yu, C. P. (1985) Theories of electrostatic lung deposition of inhaled aerosols. Ann. Occup. Hyg. 29, 219—227. Whytlaw-Gray, R. and Patterson, H. S. (1932) Smoke. Edward Arnold and Co., London. Zebel, V. G. (1958) Zur theorie des verhaltens elektrisch geladener aerosole. Kolloidzeitschrift 157, 37—50.
J. Aerosol Sci. Vol. 30, No. 4, pp. 533—548, 1999 1999 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0021-8502/99 $ — see front matter
COAGULATION OF CIGARETTE SMOKE PARTICLES R. J. Robinson and C. P. Yu* Department of Mechanical and Aerospace Engineering, State University of New York at Buffalo, Amherst, NY 14260, U.S.A. (First received 17 March 1998; and in final form 4 June 1998) Abstract—Experimental measurements on the deposition of cigarette smoke particles (CSP) in the human airways have produced results that are inconsistent with typical deposition data based on particle size. Previous work relating to hygroscopic growth indicates that hygroscopicity alone can not account for this discrepancy. The present study investigates coagulation of CSP modeled as a polydisperse-charged aerosol as a possible explanation. The results of the model more accurately predict the experimental coagulation data for mainstream CSP than models that treat CSP as a monodisperse or polydisperse-uncharged aerosol. An aerosol with an initial charge distribution based on Boltzmann equilibrium yields slightly larger coagulation rates than the mainstream CSP polydisperse-charged model. The numerical results indicate that the size and charge distribution of sidestream CSP, with a concentration of 10 particles cm\, remain stable. In 2 s, the size distribution of mainstream CSP, with a concentration of 10 particles cm\, shifts to a larger size while becoming flatter and wider. The diameter of average mass increases from 0.29 to 0.5 km. Numerical results confirm experimental reports for mainstream CSP, which indicate that the total number of charged particles increases with time and, in the early stages of coagulation, the amount of charge per particle cannot be estimated based on the particle size. This study shows that polydispersecharged CSP, allowed to coagulate for 2 s in the mouth, will not produce size distributions that yield the observed deposition of CSP. However, additional coagulation will take place as the CSP travels through the respiratory tract, which will be investigated in future work. 1999 Elsevier Science Ltd. All rights reserved
I N T RO DU CT I O N
Experimental measurements on the deposition of cigarette smoke particles (CSP) in the human respiratory tract have produced results that are inconsistent with typical deposition data. Deposition studies on monodisperse particles, other than CSP, show that for a given breathing pattern, deposition is dependent on particle size (Morrow et al., 1966a; Morrow and Yu, 1985). Fresh CSP (age 0 s) range in size from 0.04 to 0.5 km with geometric mean diameters of 0.11 km for sidestream CSP and 0.17 km for mainstream CSP (Ueno and Peters, 1986). Therefore, the expected deposition efficiency for CSP is approximately 25%. The experimental data on mainstream CSP report average deposition efficiencies of 47% (Hinds et al., 1983) and 82—97% (Mitchell, 1962; Dalhamn et al., 1968), which is higher than expected. The experimental deposition efficiencies for sidestream CSP, 7—20% (Postendorfer and Schraub, 1972; Hiller et al., 1982; Phalen et al., 1994) and 17—41% (McAughey et al., 1994), are within the range that would be expected for submicron particles 0.1—0.5 km. However, localized deposition of sidestream CSP in the tracheal bronchial airways was found to be indicative of particles in the size range 6.5—7.1 km (Phalen et al., 1994). The discrepancies between expected and experimental deposition efficiencies for CSP may be due to cloud behavior, particle charge or particle growth due to coagulation or hygroscopicity. Previous work relating to hygroscopic growth shows that a mainstream CSP with initial diameter of 0.4 km will grow to 0.6 km, and a sidestream CSP with initial diameter of 0.2 km will grow to 0.26 km (Robinson and Yu, 1998), which suggests that hygroscopicity alone cannot account for the discrepancy between expected and measured deposition
* Author to whom correspondence should be addressed. 533
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efficiencies. The present study investigates the effect of coagulation on the size and charge distribution of inhaled CSP. Monodisperse coagulation due to Brownian motion alone (Smoluchowski, 1917) predicts negligible growth during the time that CSP are inside the respiratory tract. However, experimental measurements on the coagulation rate of mainstream CSP indicate that monodisperse Brownian coagulation theory is not a satisfactory method for describing coagulation of CSP. Experimental coagulation data could not be found for sidestream CSP. Listed in Table 1 are the reported experimental coagulation constants, K, for mainstream CSP, which represent the rate of increase in the average particle volume (or mass for constant density particles) per unit time. Keith and Derrick (1960), Leonard and Kiefer (1972) and Keith (1982) determined K from the plot of the inverse concentration as a function of time. Chen et al. (1990) determined K using an empirical equation for coagulation of polydisperse aerosols which was dependent on the count median diameter. A fairly large discrepancy exists between these studies which may be due to variations in the initial size distributions and measuring techniques. With the exception of Keith and Derrick (1960), the studies listed in Table 1 report a K value significantly larger than that for Brownian monodisperse coagulation, K "8k¹C/3g, where, k is the Boltzmann’s constant,
¹ the temperature, C the slip correction and g the viscosity of the medium (Friedlander, 1977). Keith and Derrick (1960) predict a smaller growth rate than Brownian monodisperse coagulation which seems to be in error since the slowest theoretical growth rate is described by monodisperse aerosols coagulating due to Brownian motion alone. Despite the discrepancies in the K values, the studies listed in Table 1 imply that mechanisms other than Brownian motion are present during the coagulation of CSP. Davies (1988) suggests that an erroneously large coagulation rate could be due to deposition of particles by diffusion on the walls of the aging tubes. Other explanations include the effects of particle charge and polydispersity which would increase the coagulation rate. CSP is a polydisperse aerosol with standard deviations reported for sidestream CSP ranging from 1.3 to 1.7 and for mainstream CSP ranging from 1.2 to 1.5 (Okada and Matsunuma, 1974; Chang et al., 1985; Ingebrethsen, 1986; Uneo and Peters, 1986). Friedlander and Hidy (1969) applied a similarity transformation to the size distribution function for mainstream CSP to predict coagulation rates. Size distribution curves were reported and compared to those measured by Keith and Derrick (1960) for ages 30—140 s. The study concluded that the agreement between experiment and theory was fair and that the experimental measurements showed a significantly higher proportion of large particles than predicted by similarity theory. The similarity solution developed by Friedlander and Hidy (1969) assumes a self-preserving size distribution, reached after and long period of time, and would therefore not apply to the present task of predicting coagulation of inhaled CSP which would remain in the lung for less than 10 s. Furthermore, the theory predicts Brownian coagulation only and does not account for other effects such as the charge distribution.
Table 1. Experimental coagulation constants for mainstream cigarette smoke particles Investigator
K* (cm s\;10\) Reported
K (cm s\;10\)
Monodisperse
Keith and Derrick (1960) Leonard and Kiefer (1972) Keith (1982)
4.8 20.0 23.8
Chen et al. (1990)
13.3
11.0 7.5 10.8 9.7 8.6 11.0 8.5
d R (km)
t (s)
0.23 0.76 0.23 0.29 0.42 0.22 0.45
30 1.7 0.05
(d ) (d )
(d ) (d )
(MMD) (CMD) (MMD)
2.5
* Coagulation constants have been corrected based on Smoluchowski theory which defines K to be twice the slope of the line 1/N vs t plot, where N is the number concentration and t is the time. Rd "geometric mean diameter, d "diameter of the average mass, d "count mean diameter, d "mass
mean diameter, CMD"count median diameter, MMD"mass median diameter.
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It is well known that CSP contains electrically charged particles and the charge distribution of mainstream CSP has been reported in several studies (Whytlaw-Gray and Patterson, 1932; Sano et al., 1953; DallaValle et al., 1954; Hinkle et al., 1954; Holmes et al., 1959; Norman and Keith, 1965), most predicting a symmetrical charge distribution with a charge per particle of 1 or 2, and some predicating as much as 10 or more, with the number of multiple charges increasing with time. These studies are summarized in Table 2. Charge data for sidestream CSP could not be found. However, it has been postulated that the charging mechanism for mainstream smoke is chemi-ionization, due to the physical conditions of the combustion zone (Norman and Keith, 1965). Therefore, it is likely that sidestream CSP undergoes a similar ionization process and achieves a similar charge distribution. Additional charging will occur for sidestream CSP due to atmospheric ion capture. Davies (1966) shows that, in the case of symmetrically charged aerosols, the increase of the coagulation constant due to attraction will be compensated by the decrease due to repulsion. However, Davies (1966) further predicts that this compensation is only present for weak bipolar aerosols and that strongly bipolar aerosols will experience an increase in the coagulation rate. No study has been found that attempts to quantify the effect of the charge distribution on the coagulation rate of CSP. To accurately predict the growth of CSP due to coagulation, the general coagulation equation (1) must be solved for the particular charge and size distributions of CSP. The coagulation equation used in this study was first presented by Smoluchowski (1917) for Brownian motion and modified by Fuchs (1964) and Davies (1966) for charged particles. The resulting integro-differential equation cannot be solved analytically without simplifying assumptions. Zebel (1958), assuming that particle charge was unipolar and proportional to the particle volume, which eliminates the summation symbols in equation (1), presented an approximate solution for an aerosol with a trapezoidal size distribution. No study has been found that presents a solution to this equation for CSP modeled as a polydisperse aerosol with a bipolar charge distribution. The purpose of the present study was to solve the general coagulation equation, using simple numerical techniques, for CSP with a known initial size and charge distribution. The solution predicts the change in the size and charge distributions that would occur in the period that CSP might remain in the mouth before being inhaled (0—2 s). These results are required to accurately determine the size distribution of inhaled CSP. In addition, the charge distribution is required to determine the enhanced deposition of CSP in Table 2. Experimental charge distribution of mainstream cigarette smoke particles Age (min) 1—1.5
2 30—45
Charge distribution (%) Number 0 55 51 41 31 45
of charges per particle 1 2 '2 44 1.2 0 39 10 0.1 48 6.5 4.1 60 9.2 0.3 47 6.9 1.2
Number of charges per particle 0 1—10 '10 '20 46 43 12 2—5 13 65 21 2—5
Size (km)
Concentration (part/cm)
Investigator Norman and Keith (1965)
0.20 0.23 0.29 0.35 Ave
1.6 E7
Holmes et al. (1959) NR
NR
NR
Number of charges per particle 0 1—2 26 74
0.1—0.25
NR
45
Number of changes per particle 0 1 2 3 4 5 6—10 20 22 16 12 8 6 16
0.1—0.6
—
Hinke et al. (1954) Dalla Valle et al. (1954) and Whytlaw-Gray et al. (1932) Sano et al. (1953)
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the human respiratory tract due to particle charge, which has not been determined by any previous work.
TH E OR Y
The three mechanisms that affect collision rate are Brownian or thermal motion, electrostatic forces, and kinematic diffusion. The latter mechanism includes the effects of turbulence, laminar shear or velocity profile and differential sedimentation. CSP is known to be a polydisperse, charged aerosol which, when inhaled, experiences turbulent flow in the upper airways and parabolic laminar flow in the lower airways. The present study is intended to predict the initial distribution of the inhaled CSP, after coagulation in the mouth, and therefore will not include kinematic diffusion. The change in the size distribution function due to coagulation alone was first described by Smoluchowski (1917) for Brownian motion and modified by Fuchs (1964) and Davies (1966) for charged particles. The resulting equation describing the change in concentration, nS, of particles with size d and charge z in discrete form is given by I I S
1 I\ " b(z , z , d , d )n n ! b(z , z , d , d )n n , (1) T S\T G I\G T G S\T I\G T S G I T G S I 2 ! S T G T G where b is the collision frequency function. The first term on the right-hand side of equation (1) represents the increase in concentration due to collisions between particles of size d and G charge z with particles of size d and charge z , where z #z "z , and d #d "d . The T H U T U S G H I second term on the right represents the rate of decrease in concentration due to collisions between particles of size d and charge z with all other particles. The term d represents the I S I number of unit size spheres present in the agglomerate. Scanning electron photmicrographs indicate that mainstream cigarette smoke particles form clusters upon coagulation (Chen et al., 1990). The effective diameter, d , of a cluster with d size units can be determined from C I I the fractal equation (Oh and Sorensen, 1997) dnS I dt
d I d "d , I 1.3
(2)
where d is the diameter of the unit size sphere. The general form of the collision frequency function, b, neglecting kinematic diffusion is K b" , f !
(3)
in which K is the coagulation coefficient for Brownian motion and f is the ! correction factor that describes the relative effect of Coulomb forces on charged particles. The formulation of each term is discussed below. K depends on the thermal diffusion coefficients of each colliding particle and is given by (Friedlander, 1977) K
2k¹ C C G# H [d #d ], " G H d 3g d G H
(4)
where k is Boltzmann’s constant, ¹ is the absolute temperature, g is the viscosity of the medium, and C and C are the slip correction factors for particles with diameters d and G H G d , respectively. Smaller particles have larger diffusion coefficients and therefore yield H larger values of K . Polydisperse aerosols will have a larger K than mono disperse aerosols because the size distribution maximizes the two optimum conditions for collision, i.e. the larger particles provide a large surface area for collision to occur while the smaller particles have a greater thermal motion.
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537
The correction factor, f , in equation (3) due to particle charge is given by (Fuchs, ! 1964) f
(eW!1) " , ! y
(5)
where y is the ratio of thermal potential energy to electrical potential energy between the colliding particles given by 1 e y" z z , (6) 2ne k¹(d #d ) T U G H in which e is the dielectric constant of a vacuum and e is the value of an elementary charge. For attractive forces, f (1, increasing the growth rate and for repulsive forces, ! f '1, decreasing the growth rate. ! Given an initial size and charge distribution, equation (1) can be solved using simple numerical techniques to determine the size and charge distribution as a function of time for CSP particles coagulating in the mouth before being inhaled. Initial size and charge distributions are available from a number of sources as discussed in the previous section. N U ME RI CA L RE SU LT S F OR MAI NS TRE AM CSP
Equation (1) is solved numerically to determine the size and charge distributions of mainstream CSP as a function of time for a given initial size and charge distribution. The solution is used to illustrate the distinction between coagulation of mainstream CSP modeled as a polydisperse-charged aerosol and that of the simplified models for a polydisperse-uncharged or monodisperse aerosol. The models are then compared with experimental measurements made by Keith (1982) of mainstream CSP having the same initial size distribution used as the numerical input. Using the results of the polydisperse-charged model of CSP, several plots are created to illustrate the change in the size and charge distributions from 0 to 2 s in order to approximate the size and charge distribution of the inhaled CSP. These plots are also used to illustrate the relationship between particle size and particle charge. Numerical input for mainstream CSP The size distribution and concentration for fresh (age 0 s) mainstream CSP is taken from Keith (1982). As seen in Table 1, Keith (1982) provides measurements of mainstream CSP at an age of only 0.05 s. Therefore, Keith (1982) has minimized the error due to coagulation that could have taken place prior to the size measurements in the other studies listed. The method used by Keith (1982) involved the instantaneous fixation of CSP as it issued from the mouth end of the cigarette. The fixation technique used the reaction of the liquid particles with methyl 2-cyanoacrylate vapor to form a solid aerosol. Because the solid particles do not coalesce when they collide, any further coagulation after fixation could be observed. The particle size distribution of the original smoke was determined from the remaining single particles and is shown in Fig. 1 for an unfiltered cigarette. Keith (1982) estimated that the diameter measurements could be overestimated by 0.06 km. This includes 0.04 km from the 0.05 s initial aging time and 0.02 km from the addition of the methyl cyanoacrylate reactant in the particle. The apparatus was designed to minimize error due to loss of particles by wall deposition, however an estimation of this loss was not given. A total concentration of 2.86;10 particles cm\ was found for an unfiltered cigarette. The concentration was determined by counting all particles and correcting for the presence of doublets. An estimated error for the concentration was not given. The discrete size ranges used in the numerical solution are shown in Fig. 1 for mainstream CSP. For computational purposes, each size range is converted using equation (2) to represent the number of spherical size units needed to make up a particle with the given diameter. The diameter of the size unit is taken to be 0.1 km.
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R. J. Robinson and C. P. Yu
Fig. 1. Input size distribution for mainstream CSP.
The input charge distribution is based on the experimental data listed in Table 2. Complete charge data for fresh smoke could not be found in the literature. Holmes et al. (1959) made measurements after 2 minutes and did not indicate a concentration, therefore this data cannot be used to estimate an initial charge distribution. The studies by WhytlawGray and Patterson (1932), DallaValle et al. (1954) and Hinkle et al. (1954) were made soon after generation but the study did not report an exact age nor the relative percentages of particles with 1 and 2 charges. Norman and Keith (1965) listed the percent of particles having charges of 0, 1, 2 and '2 for different sizes, however, measurements were made at 1—1.5 min, although coagulation was probably minimal due to dilution of the aerosol to 10 particles cm\. Based on these studies, it was assumed that fresh mainstream CSP is made up of particles containing 0, 1 and 2 charges. Particles with charges greater than 2 were assumed not to exist in the initial distribution as indicated by the experimental data (DallaValle et al., 1954; Hinkle et al., 1954) and the conclusion made by Norman and Keith (1965) that multiple charges are developed by coagulation and not by the original burning process. Although Norman and Keith (1965) concluded that the total number of charged particles increases with diameter, their data indicate that the amount of charge per particle is not proportional to particle size. Therefore, each size range is assumed to initially contain a symmetrical charge distribution with 45% neutral particles, 47% with $1 charge and 8% with $2 charges. Comparison of charged-polydisperse model of CSP with other models The results of the numerical solution to equation (1), based on the initial size and charge distributions described above agree fairly well with the experimental results of Keith (1982). The decrease in concentration with time, calculated numerically for charged and uncharged CSP, is plotted in Fig. 2 along with the experimental results of Keith (1982) and estimations based on monodisperse coagulation with constant and varying coagulation coefficients. The results indicate that the polydisperse-charged model yields a more accurate prediction of the behavior of coagulating CSP than the simplified models of polydisperse-uncharged or monodisperse coagulation. The differences between the models are quantified in Fig. 3, a plot of inverse concentration as a function of time, from which the coagulation coefficient, K, can be determined. Regression analysis results in a second-order polynomial for both models and for the experimental data from Keith (1982), indicating that the coagulation
Coagulation of cigarette smoke particles
539
Fig. 2. Concentration with time for different models compared to experimental data for mainstream CSP.
Fig. 3. Inverse concentration with time for different models compared to experimental data for mainstream CSP.
coefficient is actually decreasing with time. However, as can be seen from the plot, K remains nearly constant, and can be approximated based on a linear best fit to the data. The K values are listed in Fig. 3 and indicate that the polydisperse-uncharged and polydisperse-charged models predict growth, respectively, of 1.6 and 1.8 times the rate of monodisperse coagulation, while Keith (1982) reported a K value of 2.7 times the rate of monodisperse coagulation. Figure 4 shows a plot of the diameter of average mass, d , as a function of time. In 1.4 s,
the numerical models predict growth ratios (d/d ), for an initial diameter d "0.29 km, of 1.52 and 1.59, respectively, for polydisperse-uncharged and polydisperse-charged models. The numerical solution for the polydisperse-charged model falls in the range of the experimental growth ratio of 1.59—1.66 estimated by Keith (1982). The lower value includes
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R. J. Robinson and C. P. Yu
Fig. 4. Diameter of average mass, d , with time for different models compared to experimental data
for mainstream CSP.
the error due to the mass of methyl cyanoacrylate reactant in the particle, as mentioned previously. Experimental and numerical models predict ratios which are larger than the 1.45 and 1.41 values predicted by monodisperse coagulation. Coagulation rates were also calculated for an aerosol with the same initial size distribution as mainstream CSP, but with an initial charge distribution based on Boltzmann charge equilibrium. A Boltzmann charge distribution is wider with a larger number of higher charged particles (Hinds, 1978) than that present in CSP. The results are included in Figs 1, 2, and 3 and indicate that a Boltzmann charge distribution yields a slightly higher coagulation rate (1.7E-9 cm s\) than charged CSP (1.54E-9 cm s\) but less than experimental measurements of charged CSP (2.38E-9 cm s\) made by Keith (1982). The results presented in this study indicate that the numerical model representing CSP as a polydisperse-charged aerosol more accurately predicts the experimental data than the simplified models of polydisperse-uncharged or monodisperse coagulation. Although the growth ratio compares well with experiment, the numerical results predict a lower coagulation coefficient than that reported by Keith (1982). The discrepancy could be due to the measuring technique. Keith (1982) varied the coagulation time using different tube lengths between the mouth end of the cigarette and the injection collar, where the particle sizes were fixed. It is possible that loss of particles due to deposition on the tube wall caused an underestimation of the measured concentration and therefore an overestimation of coagulation coefficient. This would also cause error in the size distribution, but the result could be either an overestimation or an underestimation depending on wether the loss was due to mainly to sedimentation or diffusion. No estimation of this error was provided. Another reason for the discrepancy could be an underestimation of the charge distribution. The experimental charge data listed in Table 2 indicates that larger charges could be present in CSP. More accurate experimental data for fresh tobacco smoke are needed to further investigate this possibility. Differential sedimentation could account for additional coagulation and should be considered in future studies. Size and charge distributions of inhaled mainstream CSP As a result of coagulation, the inhaled size and charge distributions will vary depending on the amount of time the CSP are held in the mouth. The size distribution frequency, based on the polydisperse-charged model, is shown in Fig. 5 for ages 0—2 s. The peak size changes from 0.2 to 0.35 km. However, d increases from 0.29 to 0.5 km, which agrees well with
Coagulation of cigarette smoke particles
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Fig. 5. Size distribution frequency for mainstream CSP modeled as a polydisperse charged aerosol for ages 0—2 s.
Keith (1982) as discussed above (Fig. 4). The results show a significant increase in the proportion of larger particles with time which agrees with the experimental results of Keith and Derrick (1960) but differs from the similarity solution presented by Friedlander and Hidy (1969), which applies only for long coagulation times. The size distribution flattens with time as the number of particles greater than 0.4 km increase and the number less than 0.4 km decrease. The results of the numerical solution presented by Zebel (1958) for a different aerosol also display this behavior of flattening with time and pivoting around a given size. Experimental data on the change in the charge distribution of CSP for times less than one minute are not available for a precise comparison. However, general trends with time and size can be observed from the studies listed in Table 2. The charge distribution frequency for CSP is plotted in Fig. 6 for ages 0—2 s. In 2 s, the proportion of neutral particles decreases from 45 to 34%, the proportion of singly charged particles remains constant at 47%, the proportion of particles with $2 charges increases from 8 to 15.5% and the proportion with $3 to $6 charges increases from 0 to 3.4%. No particles are formed with charges greater than $6 in 2 s. These results agree with measurements reported in Holmes et al. (1959) which show that the percentage of neutral particles decreases with time. Charge distribution relative to particle size is illustrated in Fig. 7. As indicated, particle charge does not increase with increasing particle size, which agrees with measurements made by Norman and Keith (1965) but differs from the assumption made by Zebel (1958). In addition, the numerical results indicate that the percent of neutral particles decreases with increasing size until 0.44 km, which agrees with Norman and Keith (1965), but reaches a minimum at 0.55 km and then increases with increasing size. Therefore, both numerical results and experimental measurements for CSP indicate that the amount of charge carried by a particle is not proportional to particle size. During the initial stages of coagulation, aerosol particles are both acquiring and losing charge as they collide with particles of the same and opposite charge, respectively. A large diameter particle could leave a larger charge range and enter a smaller charge range due to charge neutralization. Therefore, a particle’s collision history at the time of measurement, not the particle’s size, determines the amount of charge carried by the particle. Eventually, Boltzmann equilibrium is reached, in which the relationship z"2.37(d) is valid, where z is the particle charge and d is particle diameter (Hinds, 1982). For CSP in the atmosphere, equilibrium is reached in approximately 1.5 h.
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Fig. 6. Charge distribution frequency with time for polydisperse charged model of mainstream CSP.
Fig. 7. Average charge per particle with time from the polydisperse-charged model of mainstream CSP.
DallaValle et al. (1954) report that the equilibrium charge increases with diameter, although actual data for CSP was not given. The size distribution frequencies for each charge range give insight into how particle charges are formed as a result of coagulation. The size distributions for larger charges are skewed toward larger sizes, which indicates that multiple charges are formed by the process of coagulation as previously assumed in the literature (Norman and Keith, 1965). Charges not originally present (3—6) are formed immediately and their size distributions broaden and skew toward larger sizes with time. Figure 8a illustrates this trend for particles with six charges. The larger sizes could not be formed by charge neutralization because particles with charges greater than 6 were not formed in 2 s. Therefore, the larger sizes have been formed by collisions between particles of similar sign. Figure 8b shows the change in the size distribution with time for particles with three charges. The trend is the same as that for
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particles with six charges, however, the larger particles could have been formed by collisions between particles with the same or opposite charges. The peaks and valleys found in the size distribution for charge 3 were also found in measurements made by Holmes et al. (1959), Hinkle et al. (1954) and Sano et al. (1953) and could be due to the quantum nature of charge levels. The percent of singly charged particles remains unchanged (Fig. 6) and the size distribution, shown in Fig. 8c, remains narrow and shifts slowly to the right indicating that particles are entering the charge range due to charge neutralization and leaving the charge range due to collisions between charges of the same sign at approximately the same rate. A similar trend is indicated by the size distribution of neutral particles, shown in Fig. 8d, however the net effect is a loss of particles to higher charge ranges. The slow rate of formation of larger particles indicates that the neutral particles are made up primarily of the original particles
Fig. 8(a) and (b). Size distribution frequency with time from the polydisperse-charged model of mainstream CSP for charge: (a) $6, (b) $3.
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Fig. 8(c) and (d). Size distribution frequency with time from the polydisperse-charged model of mainstream CSP for charge: (c) $1, (d) 0.
in the neutral charge range and not by collisions between particles with the same but opposite charges. Three-dimensional plots of concentration as a function of size and charge are shown in Figs 9a, b and c*, for 0, 1 and 2 s, respectively. This information can be used to accurately predict the deposition of charged CSP in the human respiratory tract after coagulation in the mouth. Enhanced deposition is expected due to the increase in the average size and the presence of charged particles. Since the total concentration of particles entering the lung is on the order of 10 particles cm\, additional coagulation is expected to take place after entering the trachea. However, the magnitude of growth must be determined in conjunction with the decrease in concentration due to deposition. Furthermore, kinematic effects
* Figures 9a—c show only the distribution of the positively charged particles. An equal number of negatively charged particles exist for each charge range.
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experienced in the lung must be added to the coagulation model. These factors along with hygroscopic growth, cloud behavior and the effect of the resulting charge distribution on the deposition of CSP in the human airways will be investigated in future work using a previously developed deposition model (Chan and Yu, 1982; Yu, 1985). N U MERI CA L RE SU LT S F OR SI D ES TRE AM CSP
The sidestream CSP input size distribution was determined from the geometric mean diameter and geometric standard deviation measured by Okada and Matsunuma (1974), assuming a lognormal distribution. Each discrete size range was converted to a number of 0.1 km diameter size units using equation (2). The concentration of sidestream CSP was
Fig. 9(a) and (b). Concentration of mainstream CSP, modeled as a polydisperse-charged aerosol, at age: (a) 0 s, (b) 1 s.
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Fig. 9(c). Concentration of mainstream CSP, modeled as a polydisperse-charged aerosol, at age: (c) 2 s.
estimated to be on the order of 10 particles cm, corresponding to the maximum estimated concentration in indoor air which ranges from 20 to 3000 kg m\ (Georghlou et al., 1991). Since no charge data are available for sidestream CSP, the same charge distribution used for mainstream CSP was assumed to exist for sidestream CSP. Therefore, each size range is assumed to initially contain a symmetrical distribution with 45% neutral particles, 47% with $1 charge and 8% with $2 charges. The numerical results indicate a negligible change in the size and charge distributions of sidestream CSP aged for 2 s. The total concentration is reduced from 1.00;10 to 0.996;10 particles cm\, while the charge distribution frequency remains unchanged. The possibility of underestimating the initial strength of the charged particles must be considered. If the sidestream CSP remained in a room for 1.5 h, Boltzmann’s equilibrium would be reached at which time the charge distribution would be slightly higher. However, by this time the aerosol would be stabilized and coagulation would have ceased. Therefore, the charge distribution used in this analysis appears to be reasonable and the results indicate that the charge is not strong enough to increase the coagulation of sidestream CSP at typical concentration levels. Treating sidestream CSP as a polydisperse charged aerosol yields the same results found using monodisperse coagulation theory which predicts a stable size distribution for aerosols with concentrations on the order of 10 particles cm\ (Hinds, 1982). CON CL US I O N
The general coagulation equation (1) was solved for mainstream and sidestream CSP modeled as a polydisperse-charged aerosol. For mainstream CSP, the results more accurately represent the experimental data (Keith,1982) than models treating CSP as polydisperse-uncharged or monodisperse aerosols. Polydisperse-charged mainstream CSP was found to have a slightly smaller coagulation rate than an aerosol having an initial charge distribution representing Boltzmann equilibrium. Using the polydisperse-charged model for mainstream CSP, the size and charge distributions were determined for ages 0—2 s, representing the amount of time the CSP would
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remain in the mouth before being inhaled. In 2 s, the peak size changes from 0.2 to 0.25 km and the diameter of average mass increases from 0.29 to 0.5 km. The size distribution flattens with time and a significant number of larger particles are formed which agrees with the experimental data of Keith and Derrick (1960) but differs from the similarity solution presented by Friedlander and Hidy (1969). The change in the charge distribution of mainstream CSP predicted by the numerical model is representative of the general trends with time and particle size found in the literature. The proportion of neutral particles decreases with time, agreeing with experimental results of Holmes et al. (1959). Specifically, in 2 s, the proportion of neutral particles decreases from 45 to 34%, the proportion of singly charged particles remains constant at 47%, the proportion of particles with $2 charges increases from 8 to 15.5% and the proportion with $3 to $6 charges increases from 0 to 3.4%. No particles are formed with charges greater than $6 in 2 s. Both numerical results and experimental measurements (Norman and Keith, 1965) indicate that the number of charges carried by a particle is not proportional the particle size. Furthermore, the percent of neutral particles decreases with increasing size until 0.55 km, agreeing with Norman and Keith (1965), but then increases with increasing size. Therefore, in the early stages of coagulation, the amount of charge per particle cannot be estimated based on the particle size. The numerical results for sidestream CSP modeled as a polydisperse-charged aerosol indicate a negligible change in the size and charge distributions in 2 s. Similar results are found with monodisperse coagulation theory which predicts a stable size distribution for all aerosols with concentrations on the order of 10 particles cm\ (Hinds, 1982). The results of this study indicate that accounting for polydispersity and particle charge in the coagulation calculation does not result in sizes large enough to account for the differences in expected and measured deposition efficiencies of mainstream CSP, nor the localized deposition in the tracheal bronchial region of sidestream CSP. More accurate charge data is needed for fresh CSP to improve the accuracy of the results. Additional factors such as hygroscopic growth, kinematic coagulation and cloud effect should be considered in future studies. The charge and size distributions resulting from these numerical calculations will be used to investigate the enhanced deposition, due to particle charge, of CSP in the human airway using a previously developed deposition model (Chan and Yu, 1982; Yu, 1985). REF ER E NCE S Chan, T. L. and Yu, C. P. (1982) Charge effects on particle deposition in the human tracheobronchial tree. Ann. Occup. Hyg. 26, 65—75. Chang, P. T., Peters, L. K. and Ueno, Y. (1985) Particle size distribution of mainstream cigarette smoke undergoing dilution. Aerosol Sci. ¹echnol. 4, 191—207. Chen, B. T. Namenyi, J., Yeh, H. C., Mauderly, J. L. and Cuddihy, R. G. (1990) Physical characterization of cigarette smoke aerosol generated from a Walton smoke machine. Aerosol Sci. ¹echnol. 12, 364—375. Dalhamn, T., Edfors, M. and Rylander, R. (1968) Retention of cigarette smoke components in human lungs. Arch Environ. Health 17, 746—748. DallaValle, J. M., Orr, C. and Hinkle, B. L. (1954) The aggregation of aerosols. Br. J. Appl. Phys. 3, 198—206. Davies, C. N. (1966) Aerosol Science. pp. 42—45. Academic Press, New York. Davies, C. N. (1988) Cigarette smoke: generation and properties of the aerosol. J. Aerosol Sci. 19, 463—469. Friedlander, S. K. (1977) Smoke, Dust and Haze, pp. 175—184. Wiley, New York. Friedlander, S. K. and Hidy, G. M. (1969) New concepts in aerosol size spectrum theory. In Proc. of the 7th Int. Conf. on Condensation and Ice Nuclei (Edited by Podzimek, J.), pp. 21—25. Academea, Prague. Fuchs, N. A. (1964) ¹he Mechanics of Aerosols, pp. 305—309. Pergamon Press, Oxford. Georghiou, P. E., Blagden, P., Snow, D. A., Winsor, L. and Williams, D. T. (1991) Mutagenicity of indoor air containing environmental tobacco smoke: Evaluation of a portable pm-10 impactor sampler. Environ. Sci. ¹echnol. 25, 1496—1500. Hiller, F. C. (1984) Deposition of sidestream cigarette smoke in the human respiratory tract. Preventive Medicine 13, 602—607. Hiller, F. C., McCusker, K. T., Mazumder, M. K., Wilson, J. D. and Bone, R. C. (1982) Deposition of sidestream cigarette smoke in the human respiratory tract. Am. Rev. Respir. Dis. 125, 406—408. Hinds, W. C. (1982) Aerosol ¹echnology: Properties, Behavior, and Measurement of Airborne Particles, pp. 233—244. Wiley, New York. Hinds, W., First, M. W., Huber, G. L. and Shea, J. W. (1983) A method for measuring respiratory deposition of cigarette smoke during smoking. Am. Ind. Hyg. Assoc. J. 44, 113—118.
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Hinkle, B. L., Orr, C. Jr. and DallaValle, J. M. (1954) A new method for the measurement of aerosol electrification. J. Colloid Sci. 9, 70—80. Holmes, J. C., Hardcastle, J. E. and Mitchell, R. I. (1959) The determination of particle size and electric charge distribution in cigarette smoke. ¹obacco Science 3, 148—153. Ingebrethsen, B. J. (1986) Evolution of the particle size distribution of mainstream cigarette smoke during a puff. Aerosol Sci. ¹echnol. 5, 423—433. Keith, C. H. (1982) Particle size studies on tobacco smoke. Beitr. zur ¹abakforschung 11, 123—131. Keith, C. H. and Derrick, J. C. (1960) Measurement of the particle size distribution and concentration of cigarette smoke by conifuge. J. Colloid Science 15, 340—386. Leonard, R. E. and Kiefer, J. E. (1972) Coagulation rate of undiluted cigarette smoke. ¹obacco Science 16, 35—37. McAughey, J. J., Knight, D. A., Black, A. and Dickens, C. J. (1994) Environmental tobacco smoke retention in humans from measurements of exhaled smoke composition. Inhalation ¹oxicology 6, 615—631. Mitchell, R. I. (1962) Controlled measurement of smoke-particle retention in the respiratory tract. Am. Rev. Respr. Dis. 85, 526—533. Morrow, P. E. and Yu, C. P. (1985) Models of aerosol behavior in airways. In Aerosols in Medicine. Principles, Diagnosis and ¹herapy (Edited by Moren, F., Newhouse, M. T. and Dolovich, M. B.), pp. 149—191. Elsevier Science, New York. Norman, V. and Keith, C. H. (1965) Charged particles in cigarette smoke. ¹obacco Science 9, 75—79. Oh, C. and Sorensen, C. M. (1997) Light scattering study of fractal cluster aggregation near the free molecular regime. J. Aerosol Sci. 28, 937—957. Okada, T. and Matsunuma, K. (1974) Determination of particle size distribution and concentration of cigarette smoke by a light-scattering method. J. Colloid Interface Sci. 48, 461—469. Phalen, R. F., Oldham, M. J. and Mannix, R. C. (1994) Cigarette smoke deposition in the tracheobronchial tree: Evidence for colligative effects. Aerosol Sci. ¹echnol. 20, 215—226. Postendorfer, J. and Schraub, A. (1972) Concentration and mean particle size of the main and sidestream cigarette smoke. Staub-Reinhalt. ¸uft. 32, 33—36. Robinson, R. J. and Yu, C. P. (1998) Theoretical analysis of hygroscopic growth rate of mainstream and sidestream cigarette smoke particles in the human respiratory tract. J. Aerosol Sci. ¹echnol. 28, 21—32. Sano, K., Fujiya, Y. and Sakata, S. (1953) A colloid study of tobacco smoke: particle size distribution and electric charge. J. Jap. Chem. Soc. Pure Chem Sec. 74, 664—668. Smoluchowski, M. v. (1917) Versuch einer mathematischen theorie der koagulationskinetik kolloider losungen. Z. Physik. Chem. 92, 129—168. Ueno, Y. and Peters, L. K. (1986) Size and generation rate of sidestream cigarette smoke particles. Aerosol Sci. ¹echnol. 5, 469—476. Yu, C. P. (1985) Theories of electrostatic lung deposition of inhaled aerosols. Ann. Occup. Hyg. 29, 219—227. Whytlaw-Gray, R. and Patterson, H. S. (1932) Smoke. Edward Arnold and Co., London. Zebel, V. G. (1958) Zur theorie des verhaltens elektrisch geladener aerosole. Kolloidzeitschrift 157, 37—50.