shows the good sensitivity of this system for the determination of complement as well as the early saturation effect with increasing C’Hjo units. I t is apparent that for purposes of hemolysin titration, 5.6 C’HS0units represents a safe level of excess complement. From experimental results shown in Figures 5 and 6, it is clear that workable measurements of antibody and complement levels can be obtained by using ion-selective electrodes in conjunction with SRBC ghosts. The sensitivity of the technique is good, approaching that of the spectrophotometric method. I t is conceivable that the sensitivity may be improved by altering the concentration of SRBC ghosts. The reproducibility of our method is a function of the quality and age of the erythrocyte ghosts used. Specifically, the amounts of marker trapped and untrapped at a given time will determine the range of values obtained with increasing antibody or complement concentrations. This may vary not only with time of storage of a ghost preparation but also with the reproducibility of preparing the TMPA+ loaded erythrocytes. It is necessary, for best accuracy, to run standard curves immediately prior to unknown antibody or complement determinations. The use of ion selective electrodes for the quantitation of antibody and complement levels is attractive because electrodes can be employed in turbid biological fluids and, moreover, are capable of giving a means of monitoring such immunoreactions on a continuous basis without sampling or separations. The preparation of the necessary marker loaded SRBC ghosts is extremely simple and the resulting vesicles can be stored until needed. The experiments described here lay the ground work for further studies on immunoassay using membrane electrodes.
By using the complement fixation phenomena as an indicator reaction, it may now be possible to extend the analytical concept to a wide range of antibody determinations with the hope of good sensitivity and easy sample handling.
ACKNOWLEDGMENT We are grateful for the technical assistance of Eric J. Fogt whose thesis work served to stimulate this research. LITERATURE CITED G. A . Rechnitz, “Membrane Bioprobe Electrodes”. Chem. Eng. News, 53 (4). 29 (1975). E. A. Kaba: and M. M. Mayer, “Experimental Immunochemistry”, Charles C Thomas, Springfield, Ill., 1961, Chapter 4. G. L. Bruninga, Am. J . Clin. Pathoi., 55, 273 (1971). C. A. Aiper in “Structure and Function of Plasma Proteins”, A. C. Allison, Ed., Plenum Press, New York, N.Y., 1974, Chapter 7. M. M. Mayer, Sci. Am., 229 (5),54 (1973). G. Schwoch and H. Passow, Mol. Cell. Biochem., 2, 197 (1973). G. K. Humphries and H. M. McConneil, R o c . Natl. Acad. Sci U.S.A., 71, 1691 (1974). J. DeLoach and G. Ihler, Biochem. Biophys. Acta, 496, 136 (1977). C. M. Zmiiewski, “Immunohematoiwv”, _. Meredii CorDoration, New York, N Y , 1968, Chapter 2. M Meyerhoff and G. A. Rechnitz, Science, 195, 494 (1977). Grand Ishnd Biological Catalog, Grand Island Biological Co., Grand Island, N.Y.. 1976-77. N. Rose, P. Bigazzi, W. Bartholomew, and R. Zarco in “Methods in Immunodiagnosis”, N. Rose and P. Bigazzi, Ed., John Wiley and Sons, New York, N.Y., 1973, Chapter 2. J. E. Kent and E. H. Fife, Jr., Am. J . Trop. Med. Hyg., 12, 103 (1963). A . D. Bangham, M. W. Hill, and N. G. A. Miller, Methods Membr. Biol., 1 , 1 (1974). G. Sessa and G. Weissmann, J . Lipid Res., 9, 310 (1968). M. N. Jones, “Biological Interfaces”, Elsevier Scientific Co., New York, N.Y., 1975, Chapter 8.
RECEIVED for review July 1, 1977. Accepted September 6, 1977.
Dynamics of the Desolvating Droplet in a Laminar Flame Kuang-Pang Li Department of Chemistry, University of Florida, Gainesville, Florida 326 1 1
Aerosol droplets first undergo desolvation in flames. During desolvation, both mass and volume, i.e., density, of a droplet change continuously. The change In density has a significant effect on the course of flight of the droplet. Taking this into consideration, rigorous expressions are derived for the description of dynamlcs of the droplet.
In the previous treatment of a flame model ( I ) , we indicated that the individual behavior of droplets sprayed into an analytical flame has a profound significance on the spatial distribution of the atomic vapor which, in turn, is closely related to the optical signal observed. Since the atom production in a flame is a very complicated process, several idealized assumptions have to be introduced to make the final expression for the overall process mathematically simple and easy enough to follow. For more rigorous analysis, the dynamics of the droplets at different stages must be described more precisely. Droplets first undergo desolvation in the flame. The position of the dry aerosol particles from these droplets depends on the velocity and the course of flight of the droplets. 2086
ANALYTICAL CHEMISTRY, VOL. 49, NO. 13, NOVEMBER 1977
Two droplets of the same diameter and from the same origin may end u p in different positions when deso!vation is complete. This is to say, distribution of the dry salt particles is not only droplet-size dependent but also velocity and course dependent. As a result, the optical signal depends heavily on the dynamics of these desolvating droplets. The purpose of this report is to demonstrate a more precise way to evaluate these dynamic parameters.
DERIVATION Consider a circular stable laminar flame of uniform composition and temperature Tf (at least uniform in the area of interest). Because of thermal expansion, the flame gases have velocities in both vertical and radial directions. Because of the fact that the flame is stable, we may assume these velocities, denoted as u and u’, respectively, to be time-independent. The origin of the coordinates ( r , 0, z ) used for describing the dynamics is set at the center of the burner head. A droplet of initial diameter, do, in cm, is introduced a t (ro,0, zo) with an initial velocity uo which makes an angle 4 with the z-axis. Since no additional 0-directed force is applied to the droplet, the droplet makes no helical movement during the flight. That
is, 0 remains constant. The dynamics of motion can, therefore, be described as a two-dimensional problem with the following equations,
or
R 2= 1 - ( k / d o 2 ) t
(12)
Since the derivatives d X / d t and d Yldt can be transformed into dX/dR and dY/& by dividing the time derivatives by the factor dR/dt = - k / 2 R d o 2 ,Equations 9 and 10 are most conveniently written as
u, and u, are the instantaneous velocities of the droplet in vertical and radial directions. F, and F, are the corresponding forces acting on the droplet. They can be expressed, respectively, as (1-31,
F,
=
3nqd(~ u,) - mg
_d _Y _ _ _3AR2Y dR R 3 + a 3
where A = 12a/kpsT, u 3 = mAo/mso, and B = 2gdo2/k. The solutions to these equations are,
(4)
X = B ( R 3+ a 3 ) AS
mo - Am
=
=
mAo+ -psTd3 71
6
where C1 and C2 are integration constants, and u is a dummy parameter for integration. Introduction of the initial conditions: R = 1,X = Xo = v - uo cos 4 and Y = Yo = u ’ - uo sin @, when t = 0, gives the final expressions for velocities u, and u,:
The position of the droplet at time t can be found by
r
=
2dO2 ro - ---Ju$dR k
which gives rise to the final expressions as follow:
z=z,+ and
dt
4-
(6)
(7)
-
+ a3)A
(5)
Combination of Equation 6 with Equations 1-4 gives
d u-r -
udu (u3
c 1 ( R 3+ a 3 ) A Y = c 2 ( R 3+ a 3 ) A
At time t s later, the droplet travels a distance up in the flame and a mass, Am, of solvent is lost. The diameter of the droplet is reduced to d cm due to desolvation. If the density of the pure solvent a t the boiling temperature, T , is denoted as p s T , Am is ( r l 6 ) p s T ( d o 3- d 3 )and the mass of the droplet a t time t is
m
=o
(3)
where mg is the gravitational force, 7 is the average viscosity, and d is the instantaneous diameter. The droplet contains analyte a t a concentration of Co, mol/cm3, which may be different from the analyte concentration, C, in the original bulk solution because of partial desolvation inside the burner. Since desolvation inside the burner is not the concern of this report, we will assume the relation between Co and C is known. The total mass of the droplet a t time t = 0 is the sum of the initial masses of the analyte (mAo)and the solvent (ms’), i.e.,
m o =m A o + m s o
and
do2V k
-(l-R2)+-
2do2(v - U O COS 4 ) -s k ( l + a3)A
@3+ 1
2d02B ~ ’ ) ~ p -d ___ p s1 P b 3 + k
37177d(~’- u,) 71
mAo+ -psTd3 6
Now, if we denote the ratio d / d o as R , L; - u, as X and u ’ u, as Y , we can transform Equations 7 and 8 into Equations 9 and 10, respectively,
g
+
dt
(187)/do2PsT)RX- = R 3 + mAo/mso
()
(9)
T h e desolvation rate is known (4-9) to be linearly dependent on time, i.e.,
d2
=
do2
-
kt
(11)
d o 2u’ r=ro+---(1-R2)+ k
2do(u‘ - u o sin ( 3 ) ___ -s G 3 + k ( l + a3)A
DISCUSSION The initial velocity distribution of droplets introduced into an analytical flame depends upon a lcit of things. The burner design, the operating conditions, the viscosity of the sample solution, etc., are all important factors. These factors together with the complex flow pattern of the flame gases make the dynamic analysis a very difficult problem. Almost every flame model adopts the vertical rising assumption to avoid exceptional mathematical complications. L’vov and co-workers ANALYTICAL CHEMISTRY, VOL. 49, NO. 13, NOVEMBER 1977
2087
Table I. Velocities of Droplets (a' = 0.0025 cm) Introduced at Different Angles in a Laminar Flame with Velocities: u = 1000.0 cm/s and u' = 10.0 cm/s Q =-goc
t, ms
u z , cmis
ur,
1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25
0.00 0.16 0.31 0.46 0.59 0.72 0.84 0.95 1.05 1.15 1.23 1.31 1.38 1.44 1.50 1.54
0.00 312.7 537.1 695.1 804.2 877.8 926.1 956.9 975.9 987.2 993.6 997.0 998.7 999.5 999.8 999.9
-
cmis
100.0 - 65.6 -40.9 - 23.5 - 11.5 - 3.4 1.9 5.3 7.4 8.6 9.3 9.7 9.9 9.9 10.0 10.0
(2, 3 ) first dealt with horizontal movement of droplets, but their treatment is not rigorous. With the intention of extending the previous model to include turbulent flames as well, we try here to describe the locus of a droplet projected into the flame gas stream at any direction and speed. In the above treatment, the launching site of the droplet is not required to be at the head of the burner. This gives more freedom for experimental verification of the theory. The setup developed by Hieftje and Malmstadt (9) will be particularly useful for this purpose. By means of Equations 21 and 22, it is possible to evaluate statistically the height distribution of the dry salt particles if the size, velocity, and angular distributions of the droplets are known. Certainly, this will require a great deal of computation. Once the distribution is established, however, it will be of great value for signal enhancement in flame spectrometry and of great help in high efficiency burner design. Equations 21 and 22 may seem to be too complicated to have any practical value. Actually, if the normal operational conditions in flame spectrometry are applied, they can be greatly simplified. In the useful analytical concentration range (0.1-10 ppm), the parameter, a3 is not significant until the ratio R is very small (-0.01). This means that a3 needs to be taken into consideration only when desolvation is nearly complete. By that time the velocity of the droplet has already approached the flame gas velocity (IO). The approach of droplet velocity to the flame velocity depends on the droplet size, the rate of desolvation, and the viscosity of the gaseous medium but less depends on the angle of injection. Table I and Figure 1 give a numerical example and schematic sketch showing how rapidly a droplet of diameter, 0.0025 cm, accelerates to the flame velocity. Under the conditions cited, the droplet approaches 99.99% of the flame velocity in the first three quarters of the droplet's desolvation, Le., R = 0.25, regardless of how the droplet is introduced. Because of this, the location where the droplet is completely desolvated can be estimated by substituting u and u' for u, and u, in Equations 19 and 20, respectively,
u,, cmis
uy,
100.0 381.4 583.3 725.6 823.8 890.0 933.5 961.2 978.3 988.5 994.2 997.3 998.8 999.5 999.8 999.9
do2 ,
ro + -u
k
In a real flame, the last part of desolvation may very likely follow a different mechanism and the analyte may start vaporizing before the droplet is completely dried. To predict 2088
ANALYTICAL CHEMISTRY, VOL. 49, NO. 13, NOVEMBER 1977
cmis
uz,
0.0 3.1 5.4 7.0 8.0 8.8 9.3 9.6 9.8 9.9 9.9
cmls
u,, cmis
0.00
100.00 71.8 51.6 37.4 27.6 21.0 16.6 13.9 12.1 11.1 10.6 10.3 10.1 10.0 10.0 10.0
312.7 537.1 695.1 804.2 877.8 926.1 956.9 975.9 987.2 993.6 997.0 998.7 999.5 999.8 999.9
10.0 10.0 10.0 10.0 10.0
h
lrn
k
(= d/do)
Flgure 1. Schematic diagram showing the approach of droplet velocities toward the Rame velocities. The following values are used for simulation: U - 0.0025 cm, k = 0.0038 cm2/s ( 6 ) ,7 = 0.00074 g/cm s ( 73), p > = 0.95838 g/cm3 ( 74), and T = 2350 K
the exact location of the dry salt particle, the physicochemical processes occurring at these times must be known quantitatively. Equations 23 and 24 can be considered as reasonable approximations to the exact location only. In the early stage of the flight, u3 is insignificant compared to R3. By dropping out this factor, the integrals can be evaluated. The locus is seen to be
z = ZO
2d02 ( u - uo cos @)(1 k k(2 + 3A) + 2 2doZB R3A+2 ) + k ( 2 - 3 A ) 3A + 2 4
+ -(1do2
~ 3 A
u,
@ = +goo
Q = 0"
R
R')u -
['-
doZ
r = ro + -(1 k
u o sin
-
2d02
RZ)u'- K(2 + 3A)'"
@)(1
-
-
R3A+2)
Figure 2 is a computer simulation of the loci of droplets introduced at different angles. It is seen that if the droplet
Winefordner (12, 12) in different analytical flames. They attributed this phenomenon to fluorescence quenching by flame components and to incomplete atomization. Both reasons cannot fully explain the facts. Because, in some cases depression occurs where the temperature profile is a maximum and in other cases depression occurs in argon-separated flames but not in unseparated flames. The flame species should be similar in these flames, particularly at the flame center, while the resulting fluorescence profiles are completely different. Whether the course of flight of droplets in these flames has significant contribution to the fluorescence depression will have to be demonstrated. In addition to the translational movement, the droplet may spin around its own axis due to nonhomogeneous evaporation at the surface. The spinning motion will exert a torque on the droplet and force the droplet to take a helical path in the flame. The present model can be modified to describe such a motion also if the acting force F(r,B) is known. Dynamics of droplets in a turbulent flame can, in principle, be studied when the spatial functions of u and u’are determined.
1.0
0.8
6
$
!
0.6
E
3L 0.4
0.2
0.0 TI
Walk DlDTUTE na*
CD(0p
ff
m
r
DI
Figure 2. Loci of desolvating aerosol droplets launched at (0.2, 0) on the burner head. The conditions used for simulation are identical to those used in Figure 1
has an initial radial velocity against the flame radial velocity, u, will first go in the negative direction, Le., toward the flame axis, pass through a minimum, and then go in the positive direction, i.e., away from the flame axis. The minimum is seen to occur a t
/
1,’
\ 1/3A
The minimum is more significant for larger angle 4. As expected, no minimum in r is observed for droplets which have a radial velocity parallel to u’ initially. The loci of such droplets are parabolic with the launching site as their vertex. The occurrence of a minimum in the course of flight when u, is anti-parallel to u’ indicates that droplets, no matter how they are introduced vertically or radially, w ill tend to fly away from the flame axis. As a result, the distribution of the dry aerosol particles, hence the optical signal, would be lower a t the center of the flame. Depression in optical signal along the flame axis has recently been observed by Harraguchi and
ACKNOWLEDGMENT The author thanks Strong S. Huang for help in developing the computer program for locus simulation.
LITERATURE CITED (1) Kuang-Pang Li, Anal. Cbem., 48, 2050 (1976). (2) B. V. L’vov, L. P. Kruglikova, L. K. Polzik, and D. A. Katskov, J . Anal. Cbem. USSR. 30, 545 (1975). (3) B. V. L’vov, L. P. Kruglikova, L. K. Polzik, and 13.A. Katskov, J . Anal. Cbem. USSR, 30, 551 (1975). (4) J. M. Beer and N. A. Cahigaer, “Combustion Aerodynamics”, Applied Science Publishers, London, 1972. (5) F. A. Williams, “Combustion Theory”. Addison Wesley, Reading, Mass., 1965. (6) N. C. Clampitt and G. M. Hieftje, Anal. Cbem., 44, 1211 (1972). (7) F. A. Williams in “8th Symposium on Combustion”, Williams and Wilkins, Baltimore, Md., 1962, p 50. (8) D. 6. Spalding, in “4th Symposium on Combustion”, Williams and Wilkins, Baltimore. Md.. 1953. D 847. (9) G. M. Hieftje and H. V. Malmstadt, Anal. Chem., 40, 1860 (1968). (IO) C. 8. Boss and G. M. Hieftje, Anal. Cbem. (in press). (11) H. Haraguchi and J. D. Winefordner, Appl. Spectrosc., 31, 195 (1977). (12) H. Haraguchl and J. D. Winefordner, Appl. Spectrosc., 31, 330 (1977). (13) D. J. Poferl and R . A. Svehla, NASA Tech. Note, D-7488 (1973). (14) G. S. Kell, J . Cbem. Eng. Data, 12, 67 (1967).
RECEIVED for review June 9, 1977. Accepted September 1, 1977. Financial support from the University Computing Support, University of Florida is gratefully acknowledged.
ANALYTICAL CHEMISTRY, VOL. 49, NO. 13, NOVEMBER 1977
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