- Email: [email protected]
The use of differential mobility analyzers of second order in determining the aerosol size distribution
J. AerosolSci. Vol. 12, pp. 55 to 57.
0021-8502/81/0101-4)055 $02.00/0
Pergamon Press Ltd. 1981. Printed in Great Britain.
THE USE OF DIFFERENTIAL MOBILITY ANALYZERS OF SECOND ORDER IN DETERMINING THE AEROSOL SIZE DISTRIBUTION WLL1AM A. HOPPEL Naval Research Laboratory, Washington DC 20375, U,S.A.
(First received 6 June: and in revised form 6 July 1980)
In a recent paper Haaf (1980) describes an aspirated parallel plate condenser for obtaining the aerosol size distribution and comments that sufficiently accurate size distribution measurements in the Aitken region and at low background concentration cannot be achieved with methods used up to now. Except for the geometry of the chamber the method presented is conceptually the same as the method described in this Journal by Hoppel (1978) and the method of analysis presented in that paper for obtaining the size distribution is completely applicable to the parallel plate design introduced by Haaf (1980). It is the purpose of this note to call attention to my earlier paper and point out some of the similarities and differences in the analysis and use of the second differential chambers described by Hoppel (1978) and Haaf (1980). The chambers described by Hoppel (1978) and Haaf (1980) are referred to as "differential counters of the second order" by Tammet (1967). The designation originates from the fact that a simple aspirated condenser (for example a Gerdien chamber) measures integral properties of the mobility distribution such as conductivity and total ion count. In order to obtain the mobility distribution function two derivatives of the experimental current-voltage curve must be taken. Differential counters of first order are those in which one derivative is taken instrumentally either by measuring the current to only a segment of the electrode or by dividing the airflow such that the aerosol sample is a fraction of the total airflow. For measurements of the mobility distribution of atmospheric ions, the divided electrode chambers such as those described by Blackwood (1920), Misaki (1950), or Whipple (1960) have been preferred, whereas in aerosol work the preferred configuration has been to divide the airflow by introducing the aerosol sample through a thin annular slit around a core of filtered air free of particulates. The most successful of the first order chambers used in aerosol work is the electric aerosol analyzer developed by Whitby and Clark (1966) and discussed in numerous papers by Whitby, Liu, and their co-workers (see for example, Whitby, 1976). Second order chambers such as those under consideration here are chambers where both derivatives are taken instrumentally by dividing both the incoming air and the current to one electrode. In a second order chamber the data is directly related to the mobility distribution itself. Tammet (1967) has given a very general treatment for integral, first order, and second order chambers. Knutson and Whitby (1975) described a chamber very similar to the second order ion counter except for one important difference. Instead of measuring the current to an isolated segment of one electrode a small amount of air is removed through a segment of one electrode and transmitted to a CN chamber where the number of particles in the transmitted air can be counted. This has the advantage that small numbers of particles are more easily detected than small numbers of elementary charges carried by the particles. Liu and Pui (1974) had previously used this type of chamber as an electrostatic classifier to select out and transmit particles in a narrow size increment. There have also been a number of studies dealing with the deposition of aerosols along the axis of the central electrode in first order chambers with divided airflow. The analysis of the position of the deposition in terms 55
56
W . A . HOI'I'EL
of mobility given by Wahi and Liu (1971) is closely related to the mobility of the transmitted particles in the second order chambers. Since the analysis by Tammet did not consider airflow removal through an electrode, an analysis of the second-order chamber similar to that generally employed in aspiration ion counters was carried out by Hoppel (1978). This type of analysis is formulated in terms of the volume airflow rates and electrical capacitance and is general enough to be applied to almost any geometry. The parallel plate chamber described by Haaf is somewhat different in that the transmitted sample is taken from the midpoint along the axis of the chamber. The chamber can be viewed as two parallel plate chambers back-to-back. The analysis is then applied to either half. Since the notation in the papers by Hoppel and Haaf are considerably different, the similarities may not have been noticed by Haaf and others. It may be helpful to point out some of these similarities here. The basic equation which governs the number density of particles transmitted through the chamber when the input sample flow is equal to the output sample flow is given by Haaf's equation (25). If this expression is written out explicitly, Hoppel's equation (6) divided by the sample flow ~b2 is obtained. To see the equivalence of the two equations the midpoint mobility Z 0 of Haaf must be equal to the midpoint mobility k l of Hoppel and the mobility interval AZ of Haaf set equal to k2-kl
= kl-k3
= ((~2/~)l)kl
in Hoppel (1978). The method of deriving the basic equation in the two papers is significantly different. Hoppel derives the expression for the number of transmitted particles in terms of critical mobility analysis in Appendix A of his paper whereas Haaf expresses the number of transmitted particles as an integral containing the "transmission function". After deriving the basic equations Hoppel calculates the g-function response from his general equation and points out that this was related to the transmission function given in the Knutson-Whitby analysis. Haaf restricting his analysis to the case when the input-sample flow equals the output-sample flow, derives the transmission function necessary for his analysis. This transfer function is equal to the unit g-function response divided by the sample flow given by Hoppel. Extraction of the size distribution from the basic equation proceeds along different paths in the two papers. Hoppel calculates the mobility distribution function directly from the data using the mean-value-theorem. If the average distribution function in each interval is then plotted against the arithmetic mean the resulting mobility distribution is exact only for a distribution which is linear (or constant) over the mobility interval. In general the distribution function will not be linear across a given interval but the smaller the mobility interval (i.e. the smaller the airflow ratio ~2/q~~), the more accurate will be the approximation. Examining atmospheric data a posteriori it was found that in our case errors due to nonlinearity in the mobility distribution caused a maximum error equivalent to shifting the radius by no more than 20 ~°Fo(except in extreme cases). This means that using the mean-valuetheorem causes a maximum uncertainty of ± 20 ~o in the radius resolution for our mobility intervals. As discussed in the original paper air flows and voltages were chosen to give non overlapping mobility channels such that doubly charged particles in one channel will appear totally in another channel. This makes it possible to cover the desired range in the minimum number o f voltage steps and account exactly for particles carrying an even number of charges. Corrections for particles carrying odd numbers of elementary charges (greater than one) are only approximate because particles from one size interval carrying an odd number of charges will appear in two different mobility channels. After the determination of the mobility distribution it is converted to a size distribution assuming a Boltzmann's charge distribution. As described in the original paper (Hoppel, 1978) this is done by an iterative process which can be as exact as desired within the limitations of finite size channels dictated by the mobility data. Haaf proceeds in a somewhat different manner expressing the mobility distribution in terms of the size distribution with various charge species of different size contributing to the mobility distribution at a given mobility. The integral is then inverted numerically to find the
Mobility analyzers to determine aerosol size distribution
57
size distribution. Since we are not familiar with the exact method of inversion used we are not in a position to comment on the relative merits of the two methods. The NRL mobility analyzer has been used at NRL since 1977. In its initial configuration a long tube Pollak counter was used to detect the transmitted particles. In this configuration the instrument proved to be a valuable tool for making size distribution measurements in all but the cleanest environments (Hoppel, 1978, 1979). In 1978 a segmented vertical thermal diffusion cloud chamber was constructed to grow the transmitted particles to optically detectable sizes for single particle counting. When used in conjunction with the mobility analyzer this CN counter described in this journal (Hoppel et al., 1979), gives almost unlimited sensitivity in measuring the size distribution (Hoppel, 1980). The mobility analyzer has now been automated with the aid of a minicomputer so that real time plots of the size distribution can be obtained every 20 min. We agree with Haaf that differential chambers of second order are certainly the most accurate method of determining the size distribution of CN currently in use and compliment him on what appears to be a carefully engineered system. We are not yet ready to acknowledge that the parallel plate chamber for bipolar analysis is the optimum design. The factor of two in sensitivity which is gained by measuring particles of both polarities appears to us to be more than off-set by complications in construction and undesirable edge-effects along both side walls. We have chosen to increase sensitivity by incorporating single particle counting using the new segmented CN counter. In this manner sensitivity can be increased almost indefinitely by increasing the counting time. With single particle counting only a few seconds are required for counting at each voltage level. The main time constraint is the flushing time between voltage changes.
REFERENCES Blackwood, O. (1920) Phys. Rev. 16, 85. Haaf, W. (1980) J. Aerosol Sci. 11, 189, 201. Hoppel, W. A. (1978) J. Aerosol Sci. 9, 41. Hoppel, W. A. (1979) J. Attn. Sci. 36, 2006. Hoppel, W.A. (1980) Atmospheric Aerosols: Formation, Optical Properties and Effects. (Edited by Deepak, A.) Proceedings of Workshop sponsored by Inst. for Atmos. Optics & Remote Sensing, Spectrum Press, Hampton, VA. Hoppel, W. A., Twomey, S. and Wojciechowski, T.A. (1979) J. Aerosol Sci. 10, 369. Knutson, E. O. and Whitby, K.T. (1975) J. Aerosol Sci. 6, 443, 453. Liu, B. Y. H. and Pui, D. Y. H. (1974) J. Colloid Interface Sci. 47, 155. Tammet, H. F. (1967)The Aspiration Method For the Determination of the Atmospheric-Ion Spectra. Translated from the Russian. Available from U.S. Department of Commerce, Clearinghouse for Federal Scient. and Tech. Info., Springfield, VA 22151. Wahl, B. N. and Liu, B. Y. H. (1971) J. Colloid Interface Sci. 37, 374. Whipple, E. (1960) J. Geophys. Res. 65, 3679. Whitby, K.T. (1976) Fine Particles. [Edited by Liu, B. Y. H.) Academic Press, New York, p. 584. Whitby, K. T. and Clark, W. E. (1966) Tellus 18, 2.
0021-8502/81/0101-4)055 $02.00/0
Pergamon Press Ltd. 1981. Printed in Great Britain.
THE USE OF DIFFERENTIAL MOBILITY ANALYZERS OF SECOND ORDER IN DETERMINING THE AEROSOL SIZE DISTRIBUTION WLL1AM A. HOPPEL Naval Research Laboratory, Washington DC 20375, U,S.A.
(First received 6 June: and in revised form 6 July 1980)
In a recent paper Haaf (1980) describes an aspirated parallel plate condenser for obtaining the aerosol size distribution and comments that sufficiently accurate size distribution measurements in the Aitken region and at low background concentration cannot be achieved with methods used up to now. Except for the geometry of the chamber the method presented is conceptually the same as the method described in this Journal by Hoppel (1978) and the method of analysis presented in that paper for obtaining the size distribution is completely applicable to the parallel plate design introduced by Haaf (1980). It is the purpose of this note to call attention to my earlier paper and point out some of the similarities and differences in the analysis and use of the second differential chambers described by Hoppel (1978) and Haaf (1980). The chambers described by Hoppel (1978) and Haaf (1980) are referred to as "differential counters of the second order" by Tammet (1967). The designation originates from the fact that a simple aspirated condenser (for example a Gerdien chamber) measures integral properties of the mobility distribution such as conductivity and total ion count. In order to obtain the mobility distribution function two derivatives of the experimental current-voltage curve must be taken. Differential counters of first order are those in which one derivative is taken instrumentally either by measuring the current to only a segment of the electrode or by dividing the airflow such that the aerosol sample is a fraction of the total airflow. For measurements of the mobility distribution of atmospheric ions, the divided electrode chambers such as those described by Blackwood (1920), Misaki (1950), or Whipple (1960) have been preferred, whereas in aerosol work the preferred configuration has been to divide the airflow by introducing the aerosol sample through a thin annular slit around a core of filtered air free of particulates. The most successful of the first order chambers used in aerosol work is the electric aerosol analyzer developed by Whitby and Clark (1966) and discussed in numerous papers by Whitby, Liu, and their co-workers (see for example, Whitby, 1976). Second order chambers such as those under consideration here are chambers where both derivatives are taken instrumentally by dividing both the incoming air and the current to one electrode. In a second order chamber the data is directly related to the mobility distribution itself. Tammet (1967) has given a very general treatment for integral, first order, and second order chambers. Knutson and Whitby (1975) described a chamber very similar to the second order ion counter except for one important difference. Instead of measuring the current to an isolated segment of one electrode a small amount of air is removed through a segment of one electrode and transmitted to a CN chamber where the number of particles in the transmitted air can be counted. This has the advantage that small numbers of particles are more easily detected than small numbers of elementary charges carried by the particles. Liu and Pui (1974) had previously used this type of chamber as an electrostatic classifier to select out and transmit particles in a narrow size increment. There have also been a number of studies dealing with the deposition of aerosols along the axis of the central electrode in first order chambers with divided airflow. The analysis of the position of the deposition in terms 55
56
W . A . HOI'I'EL
of mobility given by Wahi and Liu (1971) is closely related to the mobility of the transmitted particles in the second order chambers. Since the analysis by Tammet did not consider airflow removal through an electrode, an analysis of the second-order chamber similar to that generally employed in aspiration ion counters was carried out by Hoppel (1978). This type of analysis is formulated in terms of the volume airflow rates and electrical capacitance and is general enough to be applied to almost any geometry. The parallel plate chamber described by Haaf is somewhat different in that the transmitted sample is taken from the midpoint along the axis of the chamber. The chamber can be viewed as two parallel plate chambers back-to-back. The analysis is then applied to either half. Since the notation in the papers by Hoppel and Haaf are considerably different, the similarities may not have been noticed by Haaf and others. It may be helpful to point out some of these similarities here. The basic equation which governs the number density of particles transmitted through the chamber when the input sample flow is equal to the output sample flow is given by Haaf's equation (25). If this expression is written out explicitly, Hoppel's equation (6) divided by the sample flow ~b2 is obtained. To see the equivalence of the two equations the midpoint mobility Z 0 of Haaf must be equal to the midpoint mobility k l of Hoppel and the mobility interval AZ of Haaf set equal to k2-kl
= kl-k3
= ((~2/~)l)kl
in Hoppel (1978). The method of deriving the basic equation in the two papers is significantly different. Hoppel derives the expression for the number of transmitted particles in terms of critical mobility analysis in Appendix A of his paper whereas Haaf expresses the number of transmitted particles as an integral containing the "transmission function". After deriving the basic equations Hoppel calculates the g-function response from his general equation and points out that this was related to the transmission function given in the Knutson-Whitby analysis. Haaf restricting his analysis to the case when the input-sample flow equals the output-sample flow, derives the transmission function necessary for his analysis. This transfer function is equal to the unit g-function response divided by the sample flow given by Hoppel. Extraction of the size distribution from the basic equation proceeds along different paths in the two papers. Hoppel calculates the mobility distribution function directly from the data using the mean-value-theorem. If the average distribution function in each interval is then plotted against the arithmetic mean the resulting mobility distribution is exact only for a distribution which is linear (or constant) over the mobility interval. In general the distribution function will not be linear across a given interval but the smaller the mobility interval (i.e. the smaller the airflow ratio ~2/q~~), the more accurate will be the approximation. Examining atmospheric data a posteriori it was found that in our case errors due to nonlinearity in the mobility distribution caused a maximum error equivalent to shifting the radius by no more than 20 ~°Fo(except in extreme cases). This means that using the mean-valuetheorem causes a maximum uncertainty of ± 20 ~o in the radius resolution for our mobility intervals. As discussed in the original paper air flows and voltages were chosen to give non overlapping mobility channels such that doubly charged particles in one channel will appear totally in another channel. This makes it possible to cover the desired range in the minimum number o f voltage steps and account exactly for particles carrying an even number of charges. Corrections for particles carrying odd numbers of elementary charges (greater than one) are only approximate because particles from one size interval carrying an odd number of charges will appear in two different mobility channels. After the determination of the mobility distribution it is converted to a size distribution assuming a Boltzmann's charge distribution. As described in the original paper (Hoppel, 1978) this is done by an iterative process which can be as exact as desired within the limitations of finite size channels dictated by the mobility data. Haaf proceeds in a somewhat different manner expressing the mobility distribution in terms of the size distribution with various charge species of different size contributing to the mobility distribution at a given mobility. The integral is then inverted numerically to find the
Mobility analyzers to determine aerosol size distribution
57
size distribution. Since we are not familiar with the exact method of inversion used we are not in a position to comment on the relative merits of the two methods. The NRL mobility analyzer has been used at NRL since 1977. In its initial configuration a long tube Pollak counter was used to detect the transmitted particles. In this configuration the instrument proved to be a valuable tool for making size distribution measurements in all but the cleanest environments (Hoppel, 1978, 1979). In 1978 a segmented vertical thermal diffusion cloud chamber was constructed to grow the transmitted particles to optically detectable sizes for single particle counting. When used in conjunction with the mobility analyzer this CN counter described in this journal (Hoppel et al., 1979), gives almost unlimited sensitivity in measuring the size distribution (Hoppel, 1980). The mobility analyzer has now been automated with the aid of a minicomputer so that real time plots of the size distribution can be obtained every 20 min. We agree with Haaf that differential chambers of second order are certainly the most accurate method of determining the size distribution of CN currently in use and compliment him on what appears to be a carefully engineered system. We are not yet ready to acknowledge that the parallel plate chamber for bipolar analysis is the optimum design. The factor of two in sensitivity which is gained by measuring particles of both polarities appears to us to be more than off-set by complications in construction and undesirable edge-effects along both side walls. We have chosen to increase sensitivity by incorporating single particle counting using the new segmented CN counter. In this manner sensitivity can be increased almost indefinitely by increasing the counting time. With single particle counting only a few seconds are required for counting at each voltage level. The main time constraint is the flushing time between voltage changes.
REFERENCES Blackwood, O. (1920) Phys. Rev. 16, 85. Haaf, W. (1980) J. Aerosol Sci. 11, 189, 201. Hoppel, W. A. (1978) J. Aerosol Sci. 9, 41. Hoppel, W. A. (1979) J. Attn. Sci. 36, 2006. Hoppel, W.A. (1980) Atmospheric Aerosols: Formation, Optical Properties and Effects. (Edited by Deepak, A.) Proceedings of Workshop sponsored by Inst. for Atmos. Optics & Remote Sensing, Spectrum Press, Hampton, VA. Hoppel, W. A., Twomey, S. and Wojciechowski, T.A. (1979) J. Aerosol Sci. 10, 369. Knutson, E. O. and Whitby, K.T. (1975) J. Aerosol Sci. 6, 443, 453. Liu, B. Y. H. and Pui, D. Y. H. (1974) J. Colloid Interface Sci. 47, 155. Tammet, H. F. (1967)The Aspiration Method For the Determination of the Atmospheric-Ion Spectra. Translated from the Russian. Available from U.S. Department of Commerce, Clearinghouse for Federal Scient. and Tech. Info., Springfield, VA 22151. Wahl, B. N. and Liu, B. Y. H. (1971) J. Colloid Interface Sci. 37, 374. Whipple, E. (1960) J. Geophys. Res. 65, 3679. Whitby, K.T. (1976) Fine Particles. [Edited by Liu, B. Y. H.) Academic Press, New York, p. 584. Whitby, K. T. and Clark, W. E. (1966) Tellus 18, 2.