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A calibration process for single-channel coulter particle counters
A Calibration Processfor Single-Channel Coulter Particle Counters Coulter counters (1, 2) need to be calibrated for each particular operating condition. Calibration is especially important if the instrument is to be used for the determination of particle size distributions over wide ranges of particle sizes. Some of the problems associated with the calibration of Coulter counters have been discussed by Kubitschek (3) and Mercer (4). Because instrument response is linear, at least over restricted ranges, the calibration is usually expressible in terms of a single constant of proportionality K, which relates pulse heights to particle volumes. With single-channel Coulter counters (models B, Zu among others), the procedure normally used for determining K is to find the so-called halLcount. With particles of known size, customarily monodispersed latex particles, the threshold T~ at which one-half of the total count is registered is found. This threshold in essence then corresponds to the median of the pulseheight distribution produced by the calibrating particles. The half-count determination must be done with appropriately selected aperture current and amplification settings. The selection is based on an assessment of the median pulse height from a cathode-ray tube display of pulse trains on the instrument. Choosing settings that are ]ess than optimal can lead to erroneous K values. The half-count method of calibration is simple, rapid, and usually adequate. HoweveL the method also can lead to false calibration constants in certain cases. The most obvious sources of error are: (1) the possibility that foreign particles are present in the liquid in addition to the particle standards, and (2) the presence of instrument noise. Both of these potential errors are of greatest importance when attempting to calibrate the instrument for the smallest particle sizes measurable. The presence of these errors cannot be detected if only standard procedures are followed. The calibration method to be outlined below circumvents the problems mentioned and is of more general applicability than the half-count procedure. The new method has the added advantage of being quite "transparent," i.e., the validity of the calibration can be gauged clearly from the results of the calibration run. The basic relation between pulse height, as expressed by the threshold setting T and the particle volume V is V = K1AT,
[1]
where I is the aperture current setting and A is the
amplification setting. These settings refer to what is actually indicated on the instrument; the physical quantities vary as the inverse of the settings. With the half-count technique, K is obtained by substituting V~, the volume of tile particle standard and 1"~.,the half-count threshold into [1]: K= V~/(IA%).
[2]
The generalized procedure we propose consists of using a monodispersed particle standard and determining the particle count for successively incremented threshold settings, just as would be done in measuring the size distribution of an unknown sample. For suctl a sequence, let 2¢-ii denote the count obtained (either for fixed sampling time or fixed sample volume) for lower and upper threshold settings of T¢ and T , respectively. With a single-threshold instrument Nij would stand for the difference between counts obtained at threshold settings of Ti and of Ti. In order to normalize N~i to equal increments in particle volume, define the quantity xI.'ii as ,1,~i = 5'¢,./(¢~,. - 4q),
[3j
e~i = I A T,..
[4]
where Evaluation of the calibration run consists of plotting ,I'ij as a function of ~I,. Basically, the ,l,-
E5]
Analytical methods also could be devised for finding %~k from the ,I,-@ readings but these would be cumbersome and would lose the clarity of the graphical approach. The uncertainty in K that results from the graphical technique is normally less than other errors involved in the analysis. 337
Copyright ~ 1975 by Academic Press. Inc. All rights of reproduction in any form reserved.
Journal of Colloid and lnlerface Science. Vol. 53, No. 2, November 1975
338
NOTES I0 ~
4~
& X I= I/4,A=f
I0 ~ iiiliiiiiiiiiiii!r.
~>
+
iiiiiiiiiiiiiiiiiiii!~,. ~:ii!i!~!iiiii:iii~:, .... ~iiiiii!iiiiii![iiiiiiiii;ii;!!iil, ;;iii!iiiiiiiiiiiiiiii!iii~iiiiiiiiifiiiii~
i0 ~
------o---I
~!!iii!giiiiiiiiiiiilfiiiii[iiiiiiiiiiiiiii~!i~
:~iiiiiiiiiiiiiii!iiiiiiiiiiiiii}iil}iiiiiiiiiiiii; ~" iii~ii,iiiiiii~i{iiiiiiiiiiiii~,ii;i}ilili~ii~ :ii!iiiiiiiiirliiiiiiiiiiiiiiiieii!i[iiiriiiiri:iiiiiiiiiiiiiieirliir~. ~-o---~-
I0
,o
~
+
;o
io
5'o
FIG. 1. Result of a calibration run with 2.02-urn latex particles. The quantities ~I' and e/, are defined by Eqs. [3] and [-4"],respectively, as~ak is the value from which the calibration constant K was obtained, using [-5]. An example is shown in Fig. 1 for the calibration of a Coulter Model ZB counter, equipped with a 19-#m aperture. The calibration was performed with latex particles of 2.02-urn diameter (Duke Standards Co., Lot No. 2274). From the cathode-ray tube display, it was evident that the peak in particle numbers was towards the
middle of the size range covered by the A = 1 setting and at the lower end of the size range covered by the A = 4 setting. Consequently, the complete distribution was obtained using only these two settings. First readings were taken, and the approximate position of the peak was found by noting counts in each of 10 equal threshold intervals for each A setting. Then, a
107 - ._~)e_.
-X-
IOe
%
• E 105
.=¢_
IE
104 +
I..-.
• A=J/4
,,=,
o o
O A=r 2~A=4 103
III A=~ } 0 A,4 A'I6 • A-G4
-O~
2=025 19,u.rn APERTURE
I:0354 70pro APERTURE 4Z-o
+
FO 2 ,_
A
I0 f J
I0 DIAMETER
()Jm)
Fro. 2. Size distribution of hydrosols in a rainwater sample. Two different sized apertures and several different amplifications were used. Journal of Colloid and Interface Science, Vol, 53, No, 2, November 1975
NOTES more precise determination of the peak was made by taking further counts over narrower intervals in the region of highest counts. Count rates were sufficiently low that coincidence corrections were unnecessary. In Fig. 1 horizontal bars show ,Ir~i for the intervals used. (Note that when the amplification setting is increased by a factor of 4, the qS-range corresponding to certain threshold settings will increase by a factor of 4.) The position of the peak in Fig. 1 can be seen to fall at ~ k = 15. The computed K value is thus 0.29. There is an uncertainty of about 3=0.5 in reading off ~i'p~k; this translates to a tolerance of 3=0.01 in K. Taking qsi to be dimensionless, the units of K are the same as that of V, in this case cubic micrometers. The upward swing of the curve on the left side indicates the presence of a background that diminishes toward larger particle sizes. This background varies from run to run and the hatched area shows the range over which it has been observed. The peak produced by the latex particles is superimposed on this background. It is apparent that the presence of this background imposes restrictions on the concentrations of the various size particles that can be used for calibration. The relatively slow decrease in counts beyond the peak demonstrates another feature of the distributions which can lead to erroneous K-values with the half-count technique. In the case illustrated, the long tail of the distribution [-whose origin is uncertain (4, 6)] actually outweighed the background count. Half-count determination of K yielded K = 0.26, corresponding to 4, = 17, which falls to the right of the peak in Fig. 1 (as shown by the arrow). Note that the consequences of having an incorrect value of K are twofold: (1) particle sizes will be in error, and (2) concentration per unit size range will be in error. When considering cumulative size distributions, the second error is not present. While it is not necessary, it is good practice to perform calibrations with several particle standards of differing sizes. The same K-value should be given by all. If there are small differences, the range of scatter indicates the degree of confidence that can be placed in the results.
339
A clear way to demonstrate the validity of the K-values used in constructing a size distribution is to show the matching of concentration values which were obtained using different A and I settings and, especially, using different apertures with a given sample. A sizedistribution for particles in a sample of rainwater is shown in Fig. 2. This distribution was determined using 19 and 70-~m apertures to cover the range from 0.56 to 20 ~m in particle size. The convergence of all measured values to a single curve bears out the point made above. What scatter there is in the points appears to be random rather than systematic. ACKNOWLEDGMENTS Support for this work was derived in part from Grant DES75-02515 by the Atmospheric Sciences Section of the National Science Foundation and in part from NCAR Subcontract 176-71 with the National Hail Research Experiment (managed by the National Center for Atmospheric Research and sponsored by the Weather Modification Program, Research Applications Directorate, National Science Foundation). Two anonymous referees have made some very helpful suggestions. REFERENCES 1. Coulter Electronics Inc., Hialeah~ Fla. 2. KIJmTSCHF,K, H. E., in "Methods in Microbiology" (R. W. Ribbins and J. R, Norris, Eds.), p. 593. Academic, London, 1969. 3. KVBI~SCH~K, H. E., Nature (London) 182, 234 (1958). 4. MERCER, W. B., Rev. Sci. Instr. 37, 1515 (1966). 5. PRINCEN,L. H., Rev. Sci. Instr. 37, 1416 (1966). 6. WALSTRA,P., AND OORTWIJN, H., J. Colloid Interface Sci. 29, 424 (1969). A. E. KING Gh~01a VALI
Department of A tmosplwric Science University of Wyoming Laramie, Wyoming Received May 27, 1975; accepted August 11, 1975
Journal of Colloid and InterfaceScience.Vol. 53, No. 2. November 1975
[1]
where I is the aperture current setting and A is the
amplification setting. These settings refer to what is actually indicated on the instrument; the physical quantities vary as the inverse of the settings. With the half-count technique, K is obtained by substituting V~, the volume of tile particle standard and 1"~.,the half-count threshold into [1]: K= V~/(IA%).
[2]
The generalized procedure we propose consists of using a monodispersed particle standard and determining the particle count for successively incremented threshold settings, just as would be done in measuring the size distribution of an unknown sample. For suctl a sequence, let 2¢-ii denote the count obtained (either for fixed sampling time or fixed sample volume) for lower and upper threshold settings of T¢ and T , respectively. With a single-threshold instrument Nij would stand for the difference between counts obtained at threshold settings of Ti and of Ti. In order to normalize N~i to equal increments in particle volume, define the quantity xI.'ii as ,1,~i = 5'¢,./(¢~,. - 4q),
[3j
e~i = I A T,..
[4]
where Evaluation of the calibration run consists of plotting ,I'ij as a function of ~I,. Basically, the ,l,-
E5]
Analytical methods also could be devised for finding %~k from the ,I,-@ readings but these would be cumbersome and would lose the clarity of the graphical approach. The uncertainty in K that results from the graphical technique is normally less than other errors involved in the analysis. 337
Copyright ~ 1975 by Academic Press. Inc. All rights of reproduction in any form reserved.
Journal of Colloid and lnlerface Science. Vol. 53, No. 2, November 1975
338
NOTES I0 ~
4~
& X I= I/4,A=f
I0 ~ iiiliiiiiiiiiiii!r.
~>
+
iiiiiiiiiiiiiiiiiiii!~,. ~:ii!i!~!iiiii:iii~:, .... ~iiiiii!iiiiii![iiiiiiiii;ii;!!iil, ;;iii!iiiiiiiiiiiiiiii!iii~iiiiiiiiifiiiii~
i0 ~
------o---I
~!!iii!giiiiiiiiiiiilfiiiii[iiiiiiiiiiiiiii~!i~
:~iiiiiiiiiiiiiii!iiiiiiiiiiiiii}iil}iiiiiiiiiiiii; ~" iii~ii,iiiiiii~i{iiiiiiiiiiiii~,ii;i}ilili~ii~ :ii!iiiiiiiiirliiiiiiiiiiiiiiiieii!i[iiiriiiiri:iiiiiiiiiiiiiieirliir~. ~-o---~-
I0
,o
~
+
;o
io
5'o
FIG. 1. Result of a calibration run with 2.02-urn latex particles. The quantities ~I' and e/, are defined by Eqs. [3] and [-4"],respectively, as~ak is the value from which the calibration constant K was obtained, using [-5]. An example is shown in Fig. 1 for the calibration of a Coulter Model ZB counter, equipped with a 19-#m aperture. The calibration was performed with latex particles of 2.02-urn diameter (Duke Standards Co., Lot No. 2274). From the cathode-ray tube display, it was evident that the peak in particle numbers was towards the
middle of the size range covered by the A = 1 setting and at the lower end of the size range covered by the A = 4 setting. Consequently, the complete distribution was obtained using only these two settings. First readings were taken, and the approximate position of the peak was found by noting counts in each of 10 equal threshold intervals for each A setting. Then, a
107 - ._~)e_.
-X-
IOe
%
• E 105
.=¢_
IE
104 +
I..-.
• A=J/4
,,=,
o o
O A=r 2~A=4 103
III A=~ } 0 A,4 A'I6 • A-G4
-O~
2=025 19,u.rn APERTURE
I:0354 70pro APERTURE 4Z-o
+
FO 2 ,_
A
I0 f J
I0 DIAMETER
()Jm)
Fro. 2. Size distribution of hydrosols in a rainwater sample. Two different sized apertures and several different amplifications were used. Journal of Colloid and Interface Science, Vol, 53, No, 2, November 1975
NOTES more precise determination of the peak was made by taking further counts over narrower intervals in the region of highest counts. Count rates were sufficiently low that coincidence corrections were unnecessary. In Fig. 1 horizontal bars show ,Ir~i for the intervals used. (Note that when the amplification setting is increased by a factor of 4, the qS-range corresponding to certain threshold settings will increase by a factor of 4.) The position of the peak in Fig. 1 can be seen to fall at ~ k = 15. The computed K value is thus 0.29. There is an uncertainty of about 3=0.5 in reading off ~i'p~k; this translates to a tolerance of 3=0.01 in K. Taking qsi to be dimensionless, the units of K are the same as that of V, in this case cubic micrometers. The upward swing of the curve on the left side indicates the presence of a background that diminishes toward larger particle sizes. This background varies from run to run and the hatched area shows the range over which it has been observed. The peak produced by the latex particles is superimposed on this background. It is apparent that the presence of this background imposes restrictions on the concentrations of the various size particles that can be used for calibration. The relatively slow decrease in counts beyond the peak demonstrates another feature of the distributions which can lead to erroneous K-values with the half-count technique. In the case illustrated, the long tail of the distribution [-whose origin is uncertain (4, 6)] actually outweighed the background count. Half-count determination of K yielded K = 0.26, corresponding to 4, = 17, which falls to the right of the peak in Fig. 1 (as shown by the arrow). Note that the consequences of having an incorrect value of K are twofold: (1) particle sizes will be in error, and (2) concentration per unit size range will be in error. When considering cumulative size distributions, the second error is not present. While it is not necessary, it is good practice to perform calibrations with several particle standards of differing sizes. The same K-value should be given by all. If there are small differences, the range of scatter indicates the degree of confidence that can be placed in the results.
339
A clear way to demonstrate the validity of the K-values used in constructing a size distribution is to show the matching of concentration values which were obtained using different A and I settings and, especially, using different apertures with a given sample. A sizedistribution for particles in a sample of rainwater is shown in Fig. 2. This distribution was determined using 19 and 70-~m apertures to cover the range from 0.56 to 20 ~m in particle size. The convergence of all measured values to a single curve bears out the point made above. What scatter there is in the points appears to be random rather than systematic. ACKNOWLEDGMENTS Support for this work was derived in part from Grant DES75-02515 by the Atmospheric Sciences Section of the National Science Foundation and in part from NCAR Subcontract 176-71 with the National Hail Research Experiment (managed by the National Center for Atmospheric Research and sponsored by the Weather Modification Program, Research Applications Directorate, National Science Foundation). Two anonymous referees have made some very helpful suggestions. REFERENCES 1. Coulter Electronics Inc., Hialeah~ Fla. 2. KIJmTSCHF,K, H. E., in "Methods in Microbiology" (R. W. Ribbins and J. R, Norris, Eds.), p. 593. Academic, London, 1969. 3. KVBI~SCH~K, H. E., Nature (London) 182, 234 (1958). 4. MERCER, W. B., Rev. Sci. Instr. 37, 1515 (1966). 5. PRINCEN,L. H., Rev. Sci. Instr. 37, 1416 (1966). 6. WALSTRA,P., AND OORTWIJN, H., J. Colloid Interface Sci. 29, 424 (1969). A. E. KING Gh~01a VALI
Department of A tmosplwric Science University of Wyoming Laramie, Wyoming Received May 27, 1975; accepted August 11, 1975
Journal of Colloid and InterfaceScience.Vol. 53, No. 2. November 1975