150
Langmuir 1985,1, 150-158
Unipolar Charging of Ultrafine Particles by Diffusion of Ions: Theory and Experiment S. W. Davison* Engineering Department, University of Maryland Baltimore County, Catonsville, Maryland 21228
S . Y . Hwang, J. Wang, and J. W. Gentry Chemical and Nuclear Engineering Department, University of Maryland, College Park, Maryland 20742 Received August 6,1984. In Final Form: November 15, 1984 Unipolar charge distributionson monodisperse ultrafine aerosols undergoing charging in a corona discharge are measured as a function of time. The measured values are compared with values calculated from the charging theories of Fuchs, Gentry, Hoppel, and Laframboise and Chang. The first three theories agree well with the experimental data. Particle loss in the charger was found to be significant. In order to obtain a meaningful comparison between theory and experiment it was necessary to integrate the charging models with an apparatus model to include the effects of particle loss and of radial variation in the aerosol flow and electrostatic field.
Introduction Applications. The electrical aerosol analyzer (EAA) is a device that imparts a unipolar charge to aerosol particles by means of a corona discharge and then precipitates the more electrically mobile (smaller or more highly charged) particles from the stream as it passes between oppositely charged metal plates (this part of the device is called the mobility analyzer). The particles entering the EAA and those emerging from the mobility analyzer can be counted. By changing the electric field strength in the mobility analyzer a mobility distribution of the aerosol stream can be obtained, and if the charge distribution on the aerosol is known the mobility distribution can be inverted to a size distribution. Up to the present time almost all users of this device have relied upon calibration curves distributed with it rather than direct knowledge of the charge distribution. An understanding of the mechanisms of unipolar charging and the ability to compute unipolar charge levels as a function of time could improve the flexibility of the device by allowing use of ion concentration-residence time products ( N t ) other than those tabulated and also by allowing corrections to be made for the effects of different ambient conditions (ion species effects). When a monodisperse aerosol is needed for experimentation it is frequently extracted from a polydisperse source by a differential mobility analyzer (DMA), which selects particles with mobilities between narrow limits. In this device the aerosol is given a bipolar equilibrium charge and the assumption is usually made that the number of multiply charged particles is negligible. If it is necessary to select particles of smaller than the mean size from the distribution, however, this assumption may not be valid. Therefore it is necessary to know the equilibrium bipolar charge distribution for small particles. This distribution is difficult to measure directly. Measurements of unipolar attachment rates will help to calculate it. Previous Work. Early work on aerosol charging by diffusion of unipolar ions is thoroughly summarized by White.lS2 White and the researchers he cites were interested in aerosol charging as a necessary preliminary to electrostatic precipitation, and their models and calcula-
tions are oriented toward obtaining design values for precipitator sizes and charger operating conditions. As a result one of the common simplifications adopted in early treatments is to regard the charging as a deterministic rather than a stochastic process; instead of obtaining a charge distribution for a given particle size these researchers obtain a specific charge nmber. More recently unipolar distributions have been calculated as part of the development of bipolar aerosol charging theories. Several references to such work are given in the section of this paper on charging models. Several experimental studies of bipolar equilibrium charging have been reported over the last 10 years.&lo The more recent papers report similar charge distributions from independent experiments using different aerosol materials and somewhat different techniques. For unipolar distributions on the other hand very little data are available. Kirsch and Zagnit'kol' measured the charge distribution on particles between 3 and 2000 nm undergoing unipolar charging. They did not report many data in the 18-42-nm range investigated here, and rather than reporting their results as a function of the ion concentration-time product N t they used a conductivity-time parameter. Their results are therefore difficult to compare with ours. If an ion mobility of 2.4 cm2/(Volt/s) is assumed, however, their results and those of the present work are consistent. Summary of the Present Investigation. To calculate unipolar charge levels it is necessary to obtain transient solutions of a system of differential equations. Among the parameters of the system are the ion attachment coefficients. Calculation of these coefficients requires the (3) (a) Liu, B.; Pui, D. J. Colloid Interface Sci. 1974,47,155. (b)Liu, B.; Pui, D. Aerosol Sci. 1974,6, 465. (4) Liu, B.; Pui, D. J. Colloid Interface Sci. 1974, 49, 305. (5) Kojima, H. Atmos. Enuiron. 1978, 12, 2363. (6) Adachi, M.; Okuyama, K.; Kousaka, Y.; Takahashi, T. J. Chem. Eng. Jpn. 1980, 13, 55. (7) Liu, C.; Gentry, J. J. Aerosol Sci. 1982, 13, 127. (8) Hussin, A.; Scheibel, H.; Becker, K.; Porstendorfer, J. J. Aerosol
Sci. 1983, 14, 671. (9) Reischl, G.; Scheibel, H.; Porstendorfer, J. J. Colloid Interface Sci. 1983, 91, 272.
(1) White, H. Am. Inst. Electr. Eng. Trans. 1951, 70, 1186. (2) White, H. "Industrial Electrostatic Precipitation"; Addison-Wesley Reading, MA, 1963.
0743-7463/85/2401-0150$01.50/0
(10) Koueaka, Y.; Adachi, M.; Okuyama, K.; Kitada, N.; Motouchi, T. Aerosol Sci. Technol. 1983,2, 421. (11) Kirsch, A.; Zagnit'ko, A. J. Colloid Interface Sci. 1981,80, 111.
0 1985 American Chemical Society
Langmuir, Vol. 1, No. 1, 1985 151
Unipolar Charging of Ultrafine Particles adoption of some model of the mechanism of aerosol charging. Several such models have been formulated. Four are used here.12-15 The derivations of these theories are discussed in detail elsewhere.16 Simulations of aerosol charging in accordance with these theories were used to identify parameter regions of interest. Some of the simulations are presented with the experimental data. Experiments were carried out to measure the unipolar charge level of a monodisperse aerosol as a function of size and Nt value. The aerosol was generated by vaporization and recondensation of silver, and a monodisperse stream was extracted by an electrical aerosol classifier (EAC). The monodisperse aerosol was neutralized and passed through a corona discharge at various different N t values. The charge distribution of the aerosol was measured using an EAA, a condensation nucleus counter (CNC), and an electrometer. The results of the experiments were then compared to the predictions of the theories. The model solutions previously discussed gave the ideal charge distributions (according to the various theories) coming out of the charger. In order to compare these values with the experimental data in a meaningful way it was necessary to simulate the operation of the EAA to take into account the particle loss. Both the simulations and experiments indicated that this particle loss is substantial and that failure to allow for it leads to significant error in interpretation of EAA data.
Charging Calculations Equation 1represents the system that must be solved dE'i/dt = NUj-1Fj-1 - NUiFi
(1)
in order to calculate aerosol charge as a function of time. Fi is the number density of particles carrying i unbalanced charges. N is the number density of ions. Viis the frequency of unipolar ion attachment to a particle carrying i unbalanced charges. Casting the equations in this form requires the assumption that the ion trajectory distributions around the particles depend only on the particle characteristics and the ambient conditions. In particular we assume that the trajectory distribution about a particle adjusts instantaneously when the particle gains or loses a charge. This validity of this pseudo-steady-state approach (which is generally accepted) hinges on the fact that the average interval between ion attachments for a given particle is much longer than the relaxation time required to establish a new ion trajectory distribution. Some theory of the mechanism of charging must be used to calculate the attachment probabilities (Vi). Because the particles are so small and their number density so low we can assume that they do not influence each other." We can regard N U as being the instantaneous current of ions to each aerosol particle. If the particle were large enough the current toward a particle of radius a could be calculated at any radial position r by means of the diffusion mobility equation: (12)Fuchs, N. Geofis. Pura Appl. 1963,56, 185. (13)Gentry, J. Aerosol Sci. 1972, 3, 65. (14)Hoppel, W.In "Electrical Processes in Atmospheres, Proceedings
of the Fifth International Conference on Atmospheric Electricity"; Dr. Dietrich Steinkopf Verlag: Darmstadt, West Germany, 1977. (15)Laframboise, J.; Chang, J. J. Aerosol Sci. 1977,8, 331. (16)Davison, S.;Gentry, J., paper presented in part at the Pacific Region Meeting of the Fine Particle Society, Honolulu, HI, Aug 1-5,1983. (17)Fuchs, N.;Sutugin, A. In 'Topics in Current Aerosol Research"; Hidy, G., Brock, J., Eds.; Pergamon Press: Oxford, England, 1971.
In this equation D is the ionic diffusivity and CP is the electrostatic potential between an ion and the particle. This equation is valid only if the gas can be regarded as a continuum, a condition which is commonly held to apply if Kn C 0.1 where Kn, the Knudsen number, is in this case defined to be the ratio of the ionic mean free path to the particle radius. Here we are concerned with particles of radius 0.100 pm or less. Since the mean free path of the ions is about 0.060 pm, this system does not meet the continuum criterion. If Kn > 10.0 it is assumed that ions near the particle undergo no collisions with each other or with neutral molecules. The charging process is said to take place in the free molecule or collisionless regime. The behavior of the gas ions may be analyzed by the techniques of kinetic gas theory and classical mechanics. Electric Potential. The potential difference between an ion (of one charge) and a point location of i charges at a distance r is called the Coulombic potential. Its value is given by (3)
The group 4m0 is a dimensionless constant. Its value is unity if electrostatic units are used. 6 is the magnitude of one elementary charge. By convention the sign on i is positive if the ion and particle are of the same polarity and negative otherwise. Aerosol particles are not point charges. If the particle (of radius a) is conductive, there is another contribution to the potential difference which is given by eq 4. (4) This is called the image potential. It is a fictional potential, a construct used to avoid the necessity of formulating a boundary condition at the particle surface. One can conceptualize it as being caused by a local charge induced in the particle by the close approach of an ion. It is always attractive and does not depend on the particle charge. The total potential difference between the particle and the ion is the sum of these two contributions. Because of the computational difficulties caused by the image potential, its, effect has occasionally been neglected in modeling aerosol charging processes. It has been shown, however, that this leads to unacceptable error,l' and this conclusion is further supported by the present work. A normalized potential (4 X 10-'/e2) is plotted in Figure 1. For each of three particle sizes there are three curves. The middle curve is the image potential, approaching negative infinity at the particle surface and zero far from the particle. The upper and lower curves are total potentials: the s u m of eq 3 and 4. Near the particle surface the image term dominates and the potential in either case approaches negative infinity. As the distance from the particle surface increases the total potential approaches f i l r , and the distance need only increase to about three radii for this to be a good approximation. The consequence is that at a distance of 100 nm from the particle center the potential field of a 5- and a 50-nm doubly charged particle are about the same. For bipolar potentials (I = -2) this means that an ion with low enough kinetic energy to be "trappable" at a distance of 100 nm by a 50-nm particle is also trappable by a 5-nm particle; bipolar attachment probabilities are not strong functions of particle size.
152 Langmuir, Vol. 1, No. I , 1985
Davison et al.
04, I
Particle Radius 5
nm
-
20nm a
02
00
-m
.. C a,
n0
-0 2
-0 4
&
Repelling Potential
-06
\
0.5
10
1.5
Log (R/nm)
Figure 1. Normalized ion-particle potential. The situation is different for the unipolar (I = 2) potential. To attach to a particle an ion must overcome the potential energy barrier indicated by the humps on the curves, and the humps get higher as the particle diameter decreases. Unipolar attachment coefficients are strong functions of particle size. Regimes. As noted above the free molecule solutions require that ions in the neighborhood of the particle undergo no collisions. This is strictly true only in the limit of infinite Knudsen number but is held to be a usable working assumption if Kn > 10.0. For 0.1 < Kn < 10.0 the system is said to be in the transition regime. Here the behavior of nearby ions must be analyzed by kinetic gas theory since the distances involved are not great enough for the averaging implicit in the diffusion equations to be valid, but it can no longer be assumed that the ions undergo no collisions whatsoever. This complicates the analysis considerably, and to date no exact solutions have been found. One way of handling this problem is to define a sphere of radius rl concentric with the particle. Outside the sphere it is assumed that the diffusion mobility equations are valid. Inside the sphere it is assumed that the collisionless treatment can be used. The solutions are matched at the sphere, eliminating the local ion number density n(rl). Another possibility is to incorporate the collisions into the kinetic gas theory analysis. This has a sound theoretical basis, but approximations are necessary to solve the equations. Impact Parameter and Minimum Apse Radius. In formulating the solution for the free molecule case most researchers have made use of the concepts of the impact parameter and the minimum apse radius. An ion’s impact parameter is the distance at which it would pass the center of the particle if it (the ion) continued in a straight line trajectory. If there were no potential difference between the particle and the ion so that the ion did indeed maintain such a trajectory it is clear that a collision would occur if
\
\__/’
Figure 2. Effective capture radii and ion trajectories. and only if the impact parameter (usually denoted b) were less than the particle radius (denoted a; see Figure 2). If the particle is carrying a charge there will be a potential difference between the particle and the ion. If the particle is conductive there will be an “image potential” (discussed previously) induced by the ion even if the particle carries no net charge. The ion, therefore, will not follow a straight line trajectory but will curve either toward or away from the particle. The curve is called an apse and the distance of the ion’s closest approach is called the apsidal distance or apse radius.14 The impact parameter and the apse radius can be related by the equations for the conservation of energy (eq 5 ) and angular momentum (eq 6). In these equations rl
+
0.5mC(r,)’ + 4(rl) = 0.5mc(r,J2 4(r,)
(5)
mrlc,(rl) = mrac(ra) (6) and ra refer to the radii of the limit sphere and the apse, respectively. The ion mass is denoted by m. We assume that the ion is traveling in a straight line at rl. The tangential component of the ion velocity at rl is ct(rl). At ra the entire velocity is tangential. From geometric considerations we can say C t ( T * ) = &)b/r1 (7) These equations can be manipulated to yield b2 = r,2[1 + [2/(mc2)l(4b(rl) - 4(r,))I (8) Thus the relationship between the apsidal distance and the impact parameter is determined by the particle charge (via the potential function) and the kinetic energy (via the particle speed). In some cases there will be an apparent minimum in b (denoted b,) as it varies with r,. The corresponding value of ra (denoted r,,,) is called the minimum apse radius. No smaller apse is possible for ions of that particular kinetic energy approaching particles carrying that particular charge. Closer approach results in capture of the ion by the particle. It is sometimes convenient to regard r , and b, as “effective”particle radii. An ion that does pass within r ,
Langmuir, Vol. 1, No. 1, 1985 153
Unipolar Charging of Ultrafine Particles
of the center of the particle is captured. An ion whose straight-line path would have carried it within b, of the particle center is captured. The curvature of the ion path due to the potential difference is incorporated in b,. If r, is used a further factor must be included to account for this curvature. Figure 2 shows these quantities schematically for the attracting and repelling cases. The ions following path B have impact parameters larger than b,, and therefore escape. The ions following path A are captured. Figure 2 should not be interpreted literally. Beyond rl ions are in Brownian motion. Their trajectories are drawn as straight lines to illustrate their impact parameters. By one theory (Fuchs) r, = a = b, for the attractive case. The minimum apse radius and minimum impact parameter (r, and b,) are functions of the ionic kinetic energy. If they are used in expressions for the current to the particle it is necessary to integrate over all energies or use some average value. Charging Models. The four models used in this investigation have been described at length and in detail elsewhere16and will only be summarized here. The Fuchs theory12J7was developed specifically to treat the problem of bipolar charging. The computations required to solve the model are fairly easy and may be rapidly carried out. Also, the model fits experimental bipolar charging data quite well. It has, however, two significant drawbacks. A simplification made in order to ease the model's computability (replacement of the group 2/(mc2) with 2/(3KT) in eq 8 imposes an artificial limit on the charging of small particles. While this is unlikely to cause error in bipolar charging calculations it is almost certain to do so in unipolar calculations. The model's other shortcoming is its use of the "limit sphere" concept. The limit sphere is a convenient theoretical construct but has no physical basis. Its use introduces arbitrary parameters for formulas into the model. Gentry's minimum flux radius (MFR) theory is an example of a family of models developed from first principles (the Boltzmann equation for the distribution of ion trajectories) by Brock, Gentry, and M a r l o ~ . ' ~ The J ~ ~most rigorous of this family of models is presented by Marlow and B r o ~ k It. ~is,~unfortunately, ~ ~ ~ only a formal solution because of the extremely lengthy computations required to evaluate it. The MFR theory makes the compromise of adopting what is in a sense an average ion trajectory for a given particle condition. While preserving the fundamental basis of the theory this makes it unnecessary to carry out a numerical integration over the entire trajectory spectrum. This integration requires a lengthy series of iterative computations at each step. Hoppel's theory belongs to a family of models developed by himself, Keefe, and Nolan.14,27-30These models are applications of ion-ion recombination theory. Hoppel's (18)Brock, J. J . Colloid Interface Sci. 1966,22,513. (19)Brock, J. J . Appl. Phys. 1970,41,843. (20)Brock, J. J. Colloid Interface Sci. 1972,39,418. (21)Brock, J.; Wu,M. J. Colloid Interface Sci. 1970,33,473. (22)Gentry, J.; Brock, J. J. Chem. Phys. 1967,47, 64. (23)Gentry, J. Ph.D. Dissehtion, University of Texas, Austin, TX, 1969. (24)Marlow, W.;Brock, J. J. Colloid Interface Sci. 1975,51,23. (25)Marlow, W.;Brock, J. J. Colloid Interface Sci. 1975,50,32. (26)Marlow, W.In 'Aerosol Measurement"; Lundgren, D., Harris, F., Marlow, W., Lippman, M., Clark, W., Eds.; University Presses of Florida: Gainesville, FL, 1979. (27)Keefe, D.; Nolan, P. Proc. R. Ir. Acad. Sect. A 1962,62A,43. (28)Keefe, D.; Nolan, P.; Scott, J. Proc. R. Ir. Acad. Sect. A 1968,66A, 17. (29)Nolan, P.; Kennan, E. Proc. R. Ir. Acad. Sect. A 1949,52, 171. (30)Hoppel, W.Pure Appl. Geophys. 1969,75,158.
U Figure 3. Schematic diagram of experiment: (1)gas cylinder, (2) dryer, (3) filter, (4)generator, (5) condenser, (6) TSI 3071 EAC, (7)TSI 3030 EAA, (8) TSI 3020 CNC, (9) vacuum pump, (10) humidifier, (11)hygrometer, (12) mixing chamber, (13) neutralizer, (14) three-way valve.
approach is that aerosol charging and ion-ion recombination must proceed by the same mechanism in the small particle limit. This introduces constructs not present in the other theories but also allows some parameters to be determined by relatively easy recombination experiments. This model is the most computationally cumbersome of those considered here due to the integrations necessary to average over the ion velocities to obtain a mean impact parameter from eq 8. Hoppel, like Fuchs, employs the limit sphere construct to handle the problem of the transition regime. Laframboise and Chang developed their aerosol charging model as an outgrowth of work done on current flow to spherical probes.15~31-33 Their fundamental equations are strictly valid only for particles much larger than those we are interested in, but they apply corrections to extend the applicability of their theory. The resulting expressions are the simplest to evaluate of the four models considered. At the limits of very large and very small particles they reduce to generally accepted results. In the transition regime they amount to a sort of interpolation formula. This theory is the only one of the four to make no attempt to deal with the problem of the image force. It is also the only theory that has been extended to nonspherical particles.
Experimental Section Procedure. Figure 3 shows a schematic diagram of the experimental apparatus. The aerosol is generated by passing filtered, dried air over vaporizing silver filters. The particles consist of homogeneously condensed silver. This process occurs in a Lindberg furnace operated between 550 and 800 "C. On one charge of filters the furnace can operate about 30 h. The furnace temperature must be raised toward the end of this period and the increased temperature leads to the generation of larger particles The rate of particle generation is a strong function of temperature. Another important furnace parameter is the gas residence time. Increased residence time broadens the distribution and increases the average particle size. The geometric standard deviation of the particle distribution leaving the furnace is about 1.5. The log mean particle diameter varies from about 14 to 28 nm as the temperature varies from 575 to 650 "C. The aerosol stream is then cooled. An electrical aerosol classifier is used to select a nearly monodisperse stream of particles from the output of the generator. Being aware of the fact that certain (31)Chang, J. J . Aerosol Sci. 1981,12,19. (32)Chang, J.; Laframboise, J. Phys. Fluids 1976,19,25. (33)Laframboise, J.; Parker, L. Phys. Fluids 1973,16,629.
Davison et al.
154 Langmuir, Vol. 1, No. 1, 1985
-
3 3 4 cm
c
Figure 4. Unipolar charger. source size distributions could give us multiply charged outliers in the EAC output, we checked it for monodispersity. We found that multiply charged particles were not a problem but that very small (D< 10 nm) particles were. Apparently these particles simply diffuse into the sample stream. The problem can be controlled by controlling the source size distribution. The particles leaving the classifier almost all carry one elementary charge. It would be convenient to pass these directly to the unipolar charger since the initial charge condition would then be clearly determined. It was found, however, that too large a fraction of the particles precipitated to the walls of the tubing, so it was necessary to neutralize them. This is accomplished by passing the stream by a krypton-85 source. There the particles acquire a bipolar equilibrium charge distribution so that most of them are uncharged. We assume that most of the charged particles are lost to the walls of the apparatus before entering the EAA. After the neutralizer is a mixing chamber where humidified air can be added. The experiments reported here were all carried out with dry air. After charge neutralization the aerosol enters the electrical aerosol analyzer, first passing through the corona discharge charger and then going into the analyzer section. The charger has been modified so that ion concentration may be adjusted by the experimenter. The stream that penetrates the analyzer section of the EAA is split. Most of it goes through an absolute filter and its total charge is measured by an electrometer. A small fraction is passed through a condensation nucleus counter so that both total charge rates and total counts are continuously available. The particle concentration in the CNC feed stream is kept below lo00 per cubic centimeter. This makes is possible to operate the CNC in the more accurate single-count mode. Experimental Model. Early in the experimental program it became evident that significant numbers of particles were being lost in the EAA through mechanisms other than precipitation in the mobility analyzer. Three possible loss mechanisms were considered (1) Brownian diffusion, ( 2 ) electrostatic precipitation in the charger due to the corona discharge field, (3) electrostatic precipitation in the apparatus due to an induced electric field (e.g., image force). Of these three the second was judged to be most significant. The charger is depicted in Figure 4 . In consists of a fine tungsten wire held at a potential of no more than 6 kV from ground. The wire is surrounded by a metal cylinder which is partly solid (sections 1 and 3) and consists partly of a metal screen
Not (
107
ion-sec1c.c.
1
Figure 5. Fraction of lost 18-nm particles. (section 2). This inner cylinder is held at a potential of about 0.2 kV. Surrounding the inner cylinder is another, larger cylinder which is grounded. The aerosol passes in laminar flow between these two cylinders. I t is charged by the ions emitted by the tungsten wire which penetrate the screen and move under the influence of the electric field toward the outer cylinder. In section 1 of the charger the aerosol is assumed to be uncharged and therefore unaffected by the field. In section 2 the aerosol is undergoing charging and the path of the charged particles will be affected by the field. In section 3 it is assumed that there are no ions and therefore no charging occurs. There is, however, an electric field so that migration and precipitation of the aerosol particles will occur. Equations 9-13 model the
-v*(aco/az)- NU& = 0 -V,(dC,/dz) - ZpE(dCl/dr)+ NUoCo- NUICl = 0 -V,(dC,/az) - 2Z$?(dC2/dr)+ NUICl - N U & = 0 -V2(dC3/dz)- 3Z$?(dc3/dr) + NU2C2- NU3C3= 0 -VZ(dC4/dz)- 4Z$?(dC4/8r)+ NU3C3- NU4C4 = 0
(9)
(10) (11) (12)
(13)
migration and charging of particles in the charger, where Ciis the number density of particles with i unbalanced charges, z is the axial coordinate, r is the radial coordinate, Vi is the ion attachment probability for a particle with i unbalanced charges, N is the number density of ions, V, is the axial velocity of the aerosol, and 2, is the electrical mobility of a singly charged particle. N is assumed to be constant in section 2 of the charger and zero elsewhere. Since the aerosol is assumed to be in a fully developed laminar flow pattern, V, is a function of radial but not axial position. This means that a particle's residence time in the charger will be a function of its radial position, another important dependence that is lost if flow in the apparatus is not modeled. Mean values of Nt are plotted and tabulated in this paper. The U i parameters are determined by the particular charging theory. The electric field (E)is assumed to be a function only of radial position and to stop abruptly at the end of section 3. In actuality it extends downstream from section 3 and is a function of axial as well as radial position. Rather than attempting to model this rigorously the assumption of the simpler functionality was made and an "effective length" of section 3 was calculated so as to give the best fit of calculated to measured particle loss. These effective lengths were used in all subsequent model calculations. Exper-
Langmuir, Vol. 1, No. 1, 1985 155
Unipolar Charging of Ultrafine Particles
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Table I. Percent Particle Loss in Charger (32-nm Particles) models (pptn region length, cm) lO-'Nt, Gentry Fuchs Hoppel Laframboise ion s/cm3 (4.2) (4.6) (5.5) (9.4) expt 25 25 25 25 25 1.44 22 20 22 22 22 1.31 20 18 1.19 20 20 20 17 19 18 18 18 1.08 15 19 0.96 16 15 16 14 13 13 14 12 0.83 11 0.70 11 11 11 10 9 9 9 8 12 0.58 6 7 7 7 8 0.45 4 8 5 5 5 0.33
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Figure 7. Fraction of lost 32-nm particles. imental measurements of the fraction of lost particles are shown in Figures 5-8 along with curves showing the fitted values from Gentry's theory. Fitted and measured losses are given in Table I for 32-nm particles. These are typical results. Although the lengths do not vary with Nt, they do increase with size, indicating that other loss mechanisms or charging model error may be lumped in with them.
Experimental Results Comparison of experiment with model predictions can be made with respect to the fraction of uncharged particles, the ratio of the total number of elementary charges leaving the unipolar charger to the number of particles entering,
lO-INt, ion s/cm3 1.44 1.31 1.19 1.08 0.96 0.83 0.70 0.58 0.45 0.33
Table 11. Percent Uncharged models Gentry Fuchs Hoppel Laframboise 3.8 40.6 0.6 7.0 0.9 8.7 5.0 43.8 6.5 47.0 1.4 10.7 2.0 13.0 8.2 50.2 10.7 54.1 3.1 16.1 14.0 4.7 20.1 58.3 7.4 25.5 18.7 62.3 68.3 32.0 24.8 11.4 17.8 40.5 33.0 74.0 27.8 51.2 44.0 80.2
expt 2.0 3.2 4.3 5.6 8.0 10.3 13.6 19.5 26.2 35.1
and the charge distribution of the aerosol leaving the charger. The fraction of uncharged particles is obtained experimentally as the ratio of the analyzer exit particle concentration when the precipitator voltage is at at its maximum to that when there is no voltage. In the first case the field strength is sufficient to precipitate all charged particles. In the second case all particles leaving the charger are counted. Figures 9-11 show the fraction of uncharged particles as a function of N t . For 32-nm particles the data are also tabulated (Table 11). As expected, as this product approaches zero the uncharged fraction
Davison et al.
156 Langmuir, Vol. 1, No. 1, 1985
Not (
107
ion-sec/c.c.
1 1.0
Figure 9. Fraction of uncharged 18-nm particles.
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Figure 11. Fraction of uncharged 42-nm particles.
w ~ t ,
ion s/cm3 1.44 1.31 1.19 1.08 0.96 0.83 0.70 0.58 0.45 0.33
\
I
I
1.0
2.0
Not ( 1 0 7 ion-sec1c.c. 1 Figure 10. Fraction of uncharged 32-nm particles. approaches unity. Simulated results using the four models are shown by the curves. The experimental data and the model curves lie on straight lines in the semilog plot. For the 32-nm data the slope for the experimental data is -2.5
Table 111. Charge/Number Ratio models Gentry Fuchs Hoppel Laframboise 0.98 1.00 1.00 1.00 1.00 0.99 0.96 0.91 0.84 0.73
0.86 0.86 0.85 0.84 0.82 0.78 0.73 0.67 0.58 0.47
0.79 0.80 0.80 0.80 0.79 0.77 0.74 0.69 0.62 0.52
0.35 0.35 0.34 0.33 0.31 0.30 0.27 0.24 0.20 0.16
expt 0.95 0.96 0.96 0.93 0.90 0.89 0.85 0.78 0.71 0.62
(ion s/cm3)-'. This lies between the Hoppel value of -2.3 and the Gentry value of -3.5. The Gentry model was observed to predict a lower fraction of uncharged particles for all particle sizes between 18 and 42 nm. The Laframboise and Chang model predicts a slope of -0.61 and a much higher fraction of uncharged particles. The most important result is that the three models that include the image term are qualitatively consistent with the experimental data. The charge/number ratio can be obtained by measuring the charge with the Faraday cage and the number density with the CNC. Data for 10 charging rates (expressed as N t product) are presented for the 32-nm case in Table I11 and plotted for this and other particle sizes in Figures 12-15. The maxima observed in the curves occur as a result of loss of the more heavily charged particles in the charger. If the particle loss in the charger had been neglected in the model calculations these curves would have been monotonically increasing; Figures 12-15 clearly illustrate the importance of particle loss. The qualitative agreement between the experimental results and those
Langmuir, Vol. 1, No. I , 1985 157
Unipolar Charging of Ultrafine Particles
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Figure 14. Total charge carried by 32-nm particles.
Figure 12. Total charge carried by 18-nm particles.
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theories including the image term is good. Generally the measurements fall between the predictions of the Fuchs and Gentry theories.
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Figure 15. Total charge carried by 42-nm particles.
The charge distribution on the particles can be measured by varying the voltage on the center rod of the analyzer. As the voltage is increased sharp drops in the number of particles penetrating the analyzer will be observed as charge classes are precipitated. The change in the total
158
Langmuir 1985, I , 158-161
Table IV. Fraction in Each Charge Class ( N t = 1.2 X lo' ion s/cm3: D = 32 nm) 0 charges 1 2+ charge/no. exptl 4.9 61 16 0.95 Gentry 1.4 57 21 1.00 54 16 0.86 Fuchs 10.7 Hoppel 6.5 67 7 0.80 33 0.3 0.34 Laframboise 47.0
number of elementary charges penetrating the analyzer can be measured with the electrometer. The change in the penetration can be measured with the CNC which thus gives the number of particles in the charge class. The ratio of these two measurements is a small integer giving the number of charges per particle. This method is usable only for small particles where the net charge is small. With larger charges the fractional change in mobility caused by adding or deleting one charge is too small to cause a sharp change in the CNC and electrometer outputs. The results of the measurements are given in Table IV. Again the
Gentry and Fuchs theories agree well with the experimental measurements.
Conclusion The importance of accounting for particle loss in the charger was indicated by both experiment and simulation. This loss occurs primarily as a result of the electrostatic precipitation of charged particles caused by the field produced by the corona discharge. In order to simulate this it is necessary to model the aerosol's parabolic flow and the charger field. A simplified model with one adjustable parameter was used. The parameter was fixed at the value that gave the best agreement between measured and calculated aerosol loss rates. Of the four models used those of Fuchs, Gentry, and Hoppel gave results that were qualitatively consistent with the data. The theory of Laframboise and Chang did not, perhaps due to the fact that this theory does not include the effects of the image potential.
Photochemical Behavior of Cetyltrimethylammonium Bromide Stabilized Colloidal Cadmium Sulfide: Effects of Surface Charge on Electron Transfer across the Colloid-Water Interface J. Kuczynski and J. K. Thomas* Chemistry Department, University of Notre Dame, Notre Dame, Indiana 46556 Received September 18, 1984. In Final Form: November 15, 1984 It is shown that ethylenediaminetetraacetate(EDTA)greatly increases the efficiency of photoinduced electron transfer from aqueous colloidal cadmium sulfate to methylviologen (MV2+).This effect is very apparent in colloids stabilized by cetyltrimethylammoniumbromide (CTAB), a cationic surfactant that imparts a positive change to the CdS surface. Experiments are reported that show that the crucial event in the system is the formation of a complex of EDTA and MV2+with a resultant negative charge. This complex is electrostaticallybound to the cationic CdS surface where photoinduced electron transfer occurs. Subsequent break up of the EDTA-reduced methylviologen complex leads to MV+, which is repelled away from the cationic CdS surface. The above effects are reversed on preparing a CdS colloid with a negatively charged surface by using sodium lauryl sulfate as a stabilizer. A mechanism for e- transfer is discussed which highlights the effect of surface type on the efficiency of e- transfer.
Introduction Photoredox reactions a t micellar and semiconductorelectrolyte interfaces have received considerable attention within the past d e ~ a d e . l - ~Colloidal semiconductor systems are of particular interest for several reasons. These systems exhibit rapid carrier mobility and efficient electron/hole separation, the ability to simultaneously carry out photooxidation and photoreduction reactions at their interfaces while also exhibiting large surface areas essential for favorable reaction yields. Furthermore, specific adsorption of ions and/or charged surfactants provides (1) Fender, J. H. "Membrane Mimetic Chemistry"; Wiley: New York, 1983. Turro, N.; Braum, A.; Gratzel, M. Angew. Chem., Int. Ed. Engl. 1980,19,675.Thomas, J. K. ACS Monogr. 1984,No. 181. Harbour, J. R.; Wolkow, R.; Hair, M. L. J. Phys. Chem. 1981,85,4026. (2)Kraeulter, B.; Bard, A. J. Am. Chem. SOC.1978,100,4317.Izumi, I.; Fan, F.-R. F.; Bard, A. J. J. Phys. Chem. 1981,85,218. Bard, A. J. J. Phys. Chem. 1982,86,172 and references therein. (3)Kalyanasundarum, K.; Borgarello, E.; Griitzel, M. Helu. Chem. Acta 1981,64,362. Gratzel, M.Acc. Chem. Res. 1981,14, 376 and ref-
erences therein. (4)Henglein, A. J. Phys. Chem. 1982,86,2291. (5) Nakato, Y.;Tsumura, A.; Tsubomura, M. Chem. Phys. Lett. 1982,
85. 387.
control of the surface charrge of the semiconductor particles via surface modification. Photochemical redox reactions induced by visible irradiation of colloidal semiconductor solutions have focused on TiOz since this semiconductor is stable with respect to anodic dissolution.6 However, the band gap of TiOz is relatively large (3.2 eV) which results in poor spectral response to visible radiation. For this reason CdS (band gap = 2.4 eV) has been utilized in numerous studies. Laser and luminescence studies"'-l2 have all been used to probe the nature of the interfacial electron-transfer reactions occurring at the semiconductor surface. Par(6)Frank, A. J.; Honda, K. J. Phys. Chem. 1982,86,1933. (7)Kuczynski, J. P.; Thomas, J. K. J. Phys. Chem. 1983,87,5498. (8)Ramsden, J. P.;Gratzel, M. J. Chem. Soc. Faraday Tram. I 1984, 80,919. (9)Rossetti, R.;Nakahara, S.; Porris, E. E. J. Chem. Phys. 1983,79, 1086. (10)Kuczynski, J. P.;Milosavljivic, B.; Thomas, J. K. J. Phys. Chem. 1984,88,980. (11)Henglein, A. Ber. Bumenges Phys. Chem. 1982,86,201. (12)Duonging,D.; Ramsden, J.; Grltzel, M. J. Am. Chem. SOC.1982, 104, 2977.
0743-7463/85/2401-0158$01.50/0 0 1985 American Chemical Society
