Pulse Characteristics of the Flame Ionization Aerosol Particle Analyzer H. C. Bolton Department
of Physics, Monash University, Clayton, Victtoria 3168, Australia
I. G. McWilliam Department of Applied Chemistry, Swinburne College of Technology, Hawthorn, Victoria 3122, Australia
This paper presents an analysis of the shape of the current pulse from a flame ionization detector when used to detect the current due to a burning spherical aerosol. A general account is given of the burning of a solid aerosol when controlled by inward diffusion of oxygen. The ionization produced by the burning changes the effective conductivity of the flame and we have to deal with a time-dependent resistance. The differential equation for the current pulse arising from such a circuit element is given and its solution is fully discussed. For the particular case of the diffusion-controlled burning of a solid aerosol, we give some numerical solutions which depend on the time constant of the circuit. The discussions are general and are also applied to the burning of an aerosol when this is controlled by the surface area. The results offer a method for determining the time constant of any flame ionization detector system.
RECENT EXPERIMENTS by Ohline ( I ) and by Crider and Strong ( 2 ) have shown that the flame ionization detector (3, 4 ) is a useful tool for examining aerosols. In this device, ionization due to the burning of each aerosol droplet, whether solid or liquid, produces a current pulse from the flame. It is the object of this paper to analyze the temporal variation of the current from the flame. The flame is treated as a time-dependent resistance and the differential equation for the current is established. The work of Ohline ( I ) established a simpler version of this differential equation; in the present paper, we give a more complete analysis showing the conditions under which Ohline’s equation is applicable. A solution for the differential equation is obtained when combustion of the aerosol is controlled by diffusion of oxygen to its surface; this yields the shape of the current pulse. The technique could also be applied to any source of current pulse, including photometric detector systems (5). It is assumed in the following that the ion current generated by the flame ionization detector is proportional to the mass rate of burning of the aerosol particle. Steady state studies have established that this applies to the burning of organic vapors over a concentration range of the order of lo8to 1, the upper limit depending on the nature of the combustible material (6). In general, this would correspond to a maximum ion current of between 0.1 and 1.O FA. The response speed of the flame ionization detector may depend either on the detector itself or on the amplifier and readout system. Amplifier limitations can generally be overcome by suitable circuit design, and it is therefore the (1) R. W. Ohline, ANAL.CHEM.,37, 93 (1965). (2) W. L. Crider and A. A. Strong, Rer. Sci. Instrum., 38, 1772 (1967). (3) I. G. McWilliam, Australian Patent 224,,504 (1957), and foreign equivalents. (4) H. C. Bolton and 1. G. McWilliam, Proc. Roy. Soc., London, A321, 361 (1971). ( 5 ) W. L. Crider, Rer. Sci. I m t r u m . , 39, 212 (1968). (6) I. G. McWilliam. J . Cliromntogr., 51, 391 (1970).
detector time constant which ultimately limits the response. To our knowledge, this has never been measured, but the analysis given below offers a method for its determination. DISCUSSION OF THE BURNING RATE OF AN AEROSOL
In this section we base our arguments substantially on the comprehensive review by Essenhigh and Fells ( 7 ) on the combustion of aerosols. Although their argument is for self-combustion rather than fol combustion in a flame, we find that this can be transferred readily to the flame detector. We assume that the aerosol particle is spherical and that the burning rate is controlled by diffusion of oxygen to its surface from a relatively large distance ro from the center of the particle. Strictly, this approach applies only to solid aerosols; with liquid aerosols, the burning rate is controlled by the outward diffusion of vapor from the evaporating drop to a burning radius. Godsave (8) and others ( 7 ) have shown that this leads to a functional form for the burning rate similar to that for solid aerosols. We shall confine our discussion here principally to solid aerosols. Contributions due to forced convection and turbulence (9, 10) will also be neglected, since this will affect only the numerical value of the effective diffusion coefficient. The rate of flow of oxygen per unit area in the outward direction is - D dc/ds where D is the diffusion coefficient, which for simplicity we assume to be constant throughout the burning, and c is the concentration of oxygen. The mass of oxygen flowing across any sphere of radius s is a constant, K, for an aerosol of given radius r(> R so that the value of R does not affect the current in the circuit and a response linear with concentration is obtained. Furthermore, the background resistance of the detector is very high and can be expected to satisfy Rz >> R , so that from Equation 17 we have R ’ cv R . This therefore justifies setting p s 0 and the solution of Equation 20 can then be obtained by using an integrating factor as discussed in the next section. From Equation 21, we obtain A ‘V EIR,, the resistance R , being a measure of the resistance of the flame when it is passing the current pulse. In general, p is expected to be small and one way of proceeding with Equation 20 is to expand Z(r) as a power series in p which is essentially a perturbation solution. Such a solution can readily be obtained in terms of multiple convolutions and, for a function f ( t ) which does not readily yield an analytic solution, such a power series would be one way of obtaining the solution.
The separate integrals can be expressed in terms of the incomplete gamma function y(s 1 , X)defined by Abramowitz and Stegun (15) through
+
y(s
+ 1, X ) = sd; use-‘du
We use the transformation u
=
(27)
(1 - x ’ ) / p to get
(15) M . Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions,” NBS Applied Mathematics Series, Vol. 55, US.Government Printing Office, Washington, D.C., 1964.
ANALYTICAL CHEMISTRY, VOL. 44, NO. 9, AUGUST 1972
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Table I. Calculation of Peak Maximum as a Function of Particle Diameter for T = 1.0 Io~x-YA
l/pl’z 10.0 7.07 4.47 3.16 2.24 1.41 1.0 0.71 0.45
P
0.01 0.02 0.05
0.1 0.2 0.5 1 .o 2.0 5.0 10 100
Figure 3. Effect of r/toon the pulse shape for diffusion limited burning. Solid curves = exact solution, Equations 29 and 30; broken curves = approximation solutions, Equations 31 .. and 37
We can at once select 1, which is the solution for get IAX) =
A P ~exp / ~{ ( I
0 ; we
of fixed size. Some results obtained from the approximation solutions are also shown. The position of the maximum of occurs at xm and we note in passing that this maximum occurs where 1o(xm) = A f ( x m )
(32)
which was proved by McWilliam and Bolton (26) and which follows readily from Equation 20 on putting dZ,/dt = 0. The value of xm is given by the solution of the equation
( 1 - xm)*/2=
- x ) i p } { 7 ~ 2l i p, ) Y(3i2, ( 1
for 0
=
0.32 0.10
EquaEquation tions Figure 3 34 32and 39 I,(x,)Ap1/2 0.97 9.7 0.95 6.7 0.91 4.1 0.84 (0.84) 2.65 0.75 1.68 0.56 0.79 0.40 0.40 0.25 (0.33) 0.18 0.12 (0.13) 0,055 0.067 0.021 0.0067 0.00067
Pl/V
- X > / P ) ) (29)
5 x 5 1 and ~ ( x =) ~ ( 1 exp ) { - ( x - 1)ipj
(30)
for x > 1. Equation 29 is equivalent t o the solution given by Ohline ( I ) which is expressed in terms of the error function. Equality of Expression 29 with that of Ohline is achieved by using the result (15): y ( 3 / 2 , A‘)
=
- X112exp (- X) +
erf
For p sufficiently large, we can expand the two exponentials in Equation 3 3 , perform the integration, and the right hand side is a power series in inverse powers of p . It then follows that
.)
(34)
Zo(xm)= 2Ato/3r
We retain Expression 29 as the more compact form of the result. However, for digital computation, Ohline’s solution is more convenient because of the availability of sub-routines for calculation of the error function (16). In the limit of very large p , the solution can readily be obtained by returning t o Equation 25 and replacing each exponential by unity. This gives
which, for a fixed value of p = r/t, shows that Zo(Xm) is proportional to the cube of the particle diameter. For p small, we can employ the asymptotic formula given by Abramowitz and Stegun (15),namely Y(S
+ 1, X )
= Y(S
+ 1,
a)
-
xSexp {-X}( 1 + S / X + s(s - 1)iXZ + . . .)
(35)
and Equation 29 becomes z,(x)
=
~ p 1 ’ 2 e x p{ ( I
- x)/p} x - XIip) x + p2/4(1 - x)’ + . . .] -
[ ( ( I - x)/P)l l 2 exp ( - ( I
(31)
and this agrees with the exact answer for p = 5 to within 15 %. We plot in Figure 3 the function Zo(x)given by Equations 29 and 30 using Pearson’s tables (17) for various values of p , and this represents the variation in pulse shape for a particle
(1
+ p/2(1 -
X)
( l / ~ ) exp * / ~(-1,’p] (1
+ pi2 + p 2 / 4 + . . .]I
(36)
which reduces to Z,(X)
(16) I. G. McWilliam and H. C . Bolton, ANAL.CHEM.,41, 1755 (1969); 43, 885 (1971). (17) K. Pearson, “Tables of the Incomplete Gamma Function,” Cambridge University Press, Cambridge, U.K., 1957. 1578
2
For p very large, Equation 34 reduces to
di
where
Z,(X) = (2A/3p)[1 - ( 1 - x ) ~ ’ ~ ]
2
(jj - 5p2 + . .
l,(xm) = A ( l - xm)l’*= A
ANALYTICAL CHEMISTRY, VOL. 44, NO. 9, AUGUST 1972
=
A[(1
- x)1/*+ p/2(1 - x)”2
+
p2/4(1 - x ) ~ ’ ~. - exp ( - x / p f ( 1
+ ..
+ p / 2 + p 2 / 4 + . . .]I
(37)
0.1 1.0 Particle Diameter (a)
Figure 4. Variation of pulse height with particle diameter for r = 1.0
Figure 5. Pulse shape for various diameter particles, with r = 1.0
The first, second, fourth, and fifth terms of Equation 37 are adequate to represent the curve Zo(x) for p = 0.1 and 0.2 t o within 2 % over the range 0 5 x 5 0.5. Taking just the first and fourth term of Equation 37 and differentiating to find the position of the maximum, x,, yields the equation
particle diameter as we would expect from Equation 8. For large values of p we find that the ion current maximum is proportional t o the cube of the particle diameter, as shown by Equation 34. This is the basis of the calibration used by Crider and Strong ( 2 ) and indicates that the pulse width is then independent of the particle diameter. The corresponding pulse shapes are shown in Figure 5 . For any value of p , we can determine the time constant of the sysiem in the following way. For t > t o , Equation 30 applies from which we obtain
2(1
- xm)l/* = p exp { x , / p }
(38)
For p = 0.1, the numerical solution of Equation 38 is xm = 0.282, compared with 0.285 from Figure 3 and for p = 0.2, x, = 0.408 compared with 0.43 from the graph. Equation 38 is therefore a very adequate representation in the limit of small p . We can solve Equation 38 algebraically for small enough values of p by taking the logarithm of both sides, and in the expansion of log (1 - xm) retaining only the first term t o get Xm =
+
{ ~ P / ( P 2)) In ( ~ / P \ I
This is an adequate expression for 0 5 p 5 0.2. I n the flame ionization aerosol particle analyzer, the time constant 7 is fixed but the particle size, and hence to, will vary. We can use Figure 3, and the approximations given in Equations 34 and 39 to show how the pulse height varies with particle size as the burning time becomes comparable with the time constant. We choose the time constant r to be unity. From Equation 7, we know that a
0:
tol12
a
l/pli2
To obtain a relationship between the particle diameter, a, and the pulse maximum, Zo(xm),we can use aZ,(x,)/A for the latter since, from Equations 11 and 22, A a a. The values of p and Zo(xm)/Aused for these calculations are given in Table I, and the results are shown in Figure 4. There are two limiting extremes, also shown in the figure. For small values of p , the particle burning curve is faithfully reproduced and the ion current maximum is proportional t o the
= Io
0 0 )
exp
{(lo
- t)/rj
and d~In lo(?) - -1/r dt For p large, we can neglect to in comparison with the pulse width, and the pulse shape is largely “tail” which is simply
Io
0:
exp (-t/7)
showing that the pulse width (and shape) is determined solely by the time constant and is independent of the particle diameter. Under these circumstances, the time constant can be determined directly from the time taken for the pulse to decrease to 3 7 x of its maximum height. At half height, the pulse width is 0.750 to for p small or r In 2 = 0.693 r for p large. Similar considerations apply to a Gaussian input distribution. Thus it is readily shown from Equation 8 of McWilliam and Bolton (16) that d In D(r)/dt
=
- l / r for r/u 2 ( 2 4 2
+
U/T)
More generally, the time constant could be obtained from the slope of the log-linear plot of I DS. t at large values of t. The expected current output from the flame ionization detector for a particle of a given diameter can be calculated
ANALYTICAL CHEMISTRY, VOL. 44, NO. 9, AUGUST 1972
1579
Table 11. Some Experimental Ionization Efficiency Figures for the Flame Ionization Detector Ionization efficiency Ref( X 106) Sample Combustion gas erence 0.07 CS-CShydrocarbons Hydrogen (18) 0.24 CI-CS hydrocarbons Hydrogenihelium (19) 0.26 n-hexane, n-heptane, Hydrogenlair (20) beeswax, dioctylphthalate 0.20 n-heptane Hydrogen/nitrogen (6)
from the known properties of this detector. Consider first a particle of hexadecane, 1 pm in diameter. The mass of this gram. For organic compounds, particle will be 4.1 X the ionization efficiency (number of ion pairs produced per carbon atom) of the flame ionization detector is approximately 0.2 X (Table 11) and the number of ion pairs produced by the total combustion of the hexadecane particle will therefore be approximately 3 X lo4. We would expect the burning time for a 1-pm diameter particle t o be approximately 3 psec, although there is a considerable range in the experimental results (7, Figure 1). Provided the system time constant is small compared with the burning time, the pulse shape will be as shown in Figure 3 for r = 0, i.e., Equation 10, and the total number of ions will be given by
6 X 10’8
im(l
- f/to)1/2dt= 4
X 10’8 imto
where im is the maximum ion current in amperes. Equating the area under the curve with the number of ion pairs produced, we obtain im N 3 X 10-9 A. Some increase in this, perhaps a factor of 3, might be expected because the burning time in the high temperature hydrogen flame will be shorter than that for hydrocarbon flames on which the above calculations are based. For small time constants, a n inherent limit to the ultimate sensitivity of the system is set by the random fluctuations in the background current due t o the flame ionization detector itself. The normal background current is in the order of 10-l’ A, although this can be decreased somewhat by purification of the gases fed to the detector. The expected fractional standard deviation in the current reading will be (12 X 10’8 io .)-I12 where io is the steady background ion current (21). For a time constant of 1 psec, and a background current of lo-” A, the corresponding inherent noise A. level will be approximately If the system time constant is much longer, a lower peak current will be obtained and, for T / t o large, the area under the current pulse curve is given by
Table 111. The Position of the Maximum xm of the Function C(x) given by Equation 47 and the Values of the Function at These Maxima, for Various Values of p Zo(xm)lA P
Equation 5 1, Equation 5 1, Equation 47 one term two terms 0.655 ... ... 0.521 .., . ~ . 0.337 ... ... 0.218 0.333 0.083 0.130 0.167 0.104 0.0593 0.0667 0.0567 0.0313 0.0333 0.0308 0.0161 0.0167 0.0161
xm
0.1
0.2 0.5 1 .o
2.0
5.0 10.0 20.0
0.19 0.28 0.42 0.53 0.64 0.76 0.82 0.87
The results of Crider and Strong ( 2 , Figure 5 ) suggest that the time constant of their detector was the limiting factor and that this was of the order of 1 msec. Using this value for the time constant, we obtain a peak maximum of 6 X lo-’? A. SOLUTION FOR A SURFACE-AREA-LIMITED BURNING RATE
It is of some comparative interest, and possibly of practical importance in low temperature combustion (9), to have the solution of the problem for a different burning rate. It is a straightforward application of the ideas developed above to find the solution when the rate of burning is proportional t o the surface area. This would occur when the diffusion coefficient for oxygen was very small. We get, using the symbols defined previously, dmldr
=
=
6
x
10’8
imT
(18) D. H. Desty, C. J. Geach, and A. Goldup, in “Gas Chromatography, 1960,” R. P. W. Scott, Ed., Butterworths, London, 1960, p 57. (19) J. C. Sternberg, W. S . Gallaway, and D. T. L. Jones, in “Gas Chromatography,” N. Brenner, Ed., Academic Press, New York, N.Y., 1961, p 231. (20) H. Frostling and P. H. Lindgren, J. Gas Cliromatogr.. 4, 243 (1966). (21) L. I. Schiff and R. D. Evans, Reo. Sci. Zmtrurn., 38, 456 (1963). 1580
(40)
where N is a constant, and thus drldt
=
-NIP
(41)
- t/to)
(42)
which integrates t o r
=
a(1
with to oc a. Referring again to the comments immediately preceding Equation 8, we say that the contribution to the flame conductivity is proportional to ldrnldtl; we see from Equations 40 and 42 that
ad?) = u2
+ uO(l - t/rJ*
(43)
and (44) We thus have f ( t ) = (1
6 X l O l s C ime-tl‘ dr
- N 4ar2
-
r/to)2,
(45)
and R, a l/a*, A 0: a?. The boundary condition is Zo(0) = 0. The solution is again given by Equation 24 and using f ( r ) from Equation 45 we get, putting x = tito, x ’ = t t / t o , and p = r/to,that
which integrates to
Z0(x) = A[(1 - x)’ -I- 2p(l - x
ANALYTICAL CHEMISTRY, VOL. 44, NO. 9, AUGUST 1972
(1
+ P) -
+ 2 p -t 2p’) exp { - X / P ] I
(47)
= =
circuit capacitance external to flame detector, Figures 1 and 2 flame detector capacitance, Figure 2
=C+C1 = diffusion coefficient for oxygen, Equation 1
.k! P
=
D function, defined by McWilliam and Bolton
=
applied voltage, Figures 1 and 2 function defined by Equation 10 background current in absence of aerosol particle maximum ion current, in amperes, due t o an aerosol particle current pulse due to an aerosol particle, Equation 15 solution of Equation 28 with p = 0 currents flowing in the circuit of Figure 2 constant in Equation 1 constant in Equation 3 instantaneous mass of burning particle constant in Equation 40
(16) = = =
= =
8 , g,, 8 , = = =
= =
= TitO X
Figure 6. Effect of burning
T i t o on
Plots of Equation 47 for various values of p are shown in Figure 6. The position of the maximum of this function occurs at the value of x = xmsatisfying
+
+ +
2(1 - X& 2pz = (1 2~ 2pZ)exp (-Xm/P) (48) A brief table of the numerical solutions of Equation 48 is
given in Table 111. In the same way as we obtained the approximate solution of Equation 38, we can solve Equation 48 approximately to get xm = {P/(I - PI} In ( 1 / 2 ~ ) (49) which is adequate for 0 2 p 2 0.2. We can get an analytic expression for the value of Zo(xm) as a function of p for large p by expanding the exponential function in Equation 48 and noting that we then have an expansion in inverse powers of p. The equation resulting from this is A(l - x,)*
=
Xm
-(xm2 3P
- 3x,
+ 3) -
(z)?
(xm2- 4x,
+ 6) + . . .
(50)
and the left hand side is Zo(xm). By assuming an expansion of xmin inverse powers of p 1 j Z it, then follows that
LIST OF SYMBOLS a
=
A
= =
C
instantaneous particle radius, Equation 2 radius representing position of source of oxygen, Equation 2 = series resistance in flame detector circuit, Figures 1 and 2 = flame resistance at pulse maximum, i.e., at t = 0, Equation 9 = effective resistance of flame, Equation 9 = flame resistance in the absence of aerosol particles, Equation 9 = effective resistance, defined by Equation 17 = radial distance measured from center of aerosol, Equation 1 = time = particle burning time, Equation 7 = time in Equation 24 =
the pulse for surface area limited
initial particle radius, Equation 6 constant, defined by Equation 21 oxygen concentration, Equation 1
=
= t/to = t'/to
value of x corresponding tc the maximum of Zo(x>,Equation 32 = incomplete gamma function, Equation 27 = integrating factor defined by Equation 23 = R t / R o ,Equation 19 = particle density, Equation 3 = standard deviation of Gaussian distribution = flame conductivity at pulse maximum, Equation 8 = flame conductivity due to burning aerosol particle, Equation 8 = constant background conductivity of flame, Equation 8 = time constant, C2Rt,Equation 18 =
RECEIVED for review November 1 1 , 1971. Accepted February 1 1 , 1972.
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