Theoretical treatment of a model for analytical flame spectrometry

Theoretical Treatment of a Model for Analytical Flame Spectrometry Kuang-pang Li Department of Chemistry, University of Florida, Gainesvilie, Ha. 326 1 1

By means of a stochastic approach, the overall processes of droplet formation and atom production are described via theoretical expressions. The atom concentration expressions describe the atomic concentration for a certain posltion in the flame as a function of solution flow rate and droplet distribution, flame gas flow velocity, atom diffusion coefficient, atomization efficiency, mass of the anaiyte atoms, solvent desolvation and solute vaporlzatlon rates, and the size of the burner. The atom concentrations can be then utilized to evaluate the signals (radiance emitted, absorbed, or fluoresced) In atomic emission, atomic absorption, and atomic fluorescence spectrometry.

Although much research has been performed on isolated processes involved in sample transformation into an optical signal (emission, absorption, or fluorescence), there have been no reports in which the analyte concentration is related via a consistent theoretical approach to the optical signal. The different stages of sample transformation into an optical signal and the processes that are important are shown in Figure 1 and have been discussed in detail by various authors. Of these authors, Alkemade ( I ) has given the best overall discussion of the processes of sample decomposition, atomization, and excitation. The individual processes of solvent desolvation, solute vaporization, compound dissociation, atom ionization, etc., are discussed by Alkemade ( I ) extremely well, i.e., there was no attempt to give a unified model relating the concentration of analyte sprayed into the flame as droplets and the resulting spatial distribution of atomic vapor in the flame and ultimately the atomic emission, absorption, or fluorescence signals to be expected. Recently, L’vov and co-workers (2, 3 ) have developed a theory describing the distribution of the analyte in flames of slotted burners. They solved the vapor transport equation directly on the basis that the atom source can be approximated as endless thin threads which are parallel to the burner slot and are situated at a height of the zone of maximum element vaporization. However, since the droplets entering the flame have a distribution of size, they will not vaporize a t the same height. The depth of the atom source will be significantly large and depends on a large number of experimental parameters. In the present report we will not attempt to review either the numerous and often complex theories of solvent evaporation in chambers and in flames and of solid particle vaporization in flames or the rather well-accepted theories of atom excitation-de-excitation processes. We will attempt to give a stochastic approach to the overall process of atom production. In this approach, a droplet is arbitrarily selected, its course in the flame is followed closely, and the analyte concentration at any observation point in the flame due to this droplet is estimated. The total analyte concentration at that point is then obtained by summing statistically the contributions from all droplets in the flame cell. The resulting ex2050

pression, if combined with the expressions for the AA, AE, or AF signals, will give spatial information about the flame cell. This theoretical model provides a novel way of thinking about the analytical flame and may be useful in many flame diagnostic studies.

THEORETICAL TREATMENT Assumptions. In this report, we will assume: (i) the analytical flame is a circular, premixed, laminar flame and the cross-section of the inner flame is essentially the same as the burner top having a radius of Ro (The choice of a circular flame is purely for mathematical convenience; a slot burner flame may be treated in a similar way); (ii) the analytical flame is uniform (constant) in temperature and composition over the region of interest and also has an outer sheath flame around the analytical flame to minimize edge effects; (iii) droplets of analyte-containing solution are produced with an efficiency of nebulization, t, (the ratio of the amount of analyte entering the flame to the amount of analyte aspirated), and the drop size distribution, P(do),is a function of the droplet diameter at the burner top, do, only; (iv) droplets enter the flame at angles not differing greatly from 90° to the burner surface; (v) the droplets are spherical and evaporate in a uniform manner producing spherical particles which vaporize in a uniform manner, that is, no explosions within droplets or particles occur; (vi) the droplets and salt particles travel upward with the speed of rising flame gases, u , which is assumed to be the same a t all points within the inner flame. A more rigorous treatment shows that the analyte rising speed will be (see Appendix)

(-

exp 2) (A-4) 1811 d02P where 7 is the viscosity of the flame gases, p is the density, and d is the diameter of the droplet or the salt particle a t time t , g is the gravity. (vii) There are no interactions between submicroscopic species (atoms, molecules, and ions) with particles or droplets, no interactions between particles and particles or droplets, and no interaction between droplets and droplets. It should be pointed out that all of these assumptions are necessary only to simplify the initial approach and are not limitations of the theoretical model, but rather are limitations of the present state of knowledge of the physical processes in flames. Description of the Stochastic Approach. Analyte solution of concentration CO,mol ~ m - is~ consumed , at a solution flow rate, F,, cm3 s-l, by a nebulizer within a chamber of a premixed flame burner system. Within the chamber, the large droplets are lost. Only a small fraction of the spray enters the flame gases in the form of tiny droplets. At any point on the burner surface, there are P(do)Gdo such tiny droplets having diameters between do and do + ado cm. Since the analytical flame is assumed to be in a steady state, these droplets are

ANALYTICAL CHEMISTRY, VOL. 48, NO. 14, DECEMBER 1976

-(u-uo--

Analyte Solution, C

Transport, Nebulization F

En 9 P (dol

Submicrostopic

-

Wet

Solvent

-

Aerosol

Evaporation

Dry '

Aerosol

Solute

3

Vaporization

Species Atoms,Molecules,

I ons

"11

I

Atoms

A€ S

AAS

Light Emission AFS

Figure 1. Pictorial model of nebulization, solvent evaporation, solute vaporization, atomization, excitation, and emission processes in flames (or other plasmas)

produced at a time-independent rate. Two stochastic approaches can be used to describe an analytical flame. In the first approach, one may follow the course of a single unique droplet (or salt particle) and solve for the transient distribution of analyte atoms produced from it. The time dependence of the distribution is then removed by means of statistical methods. Or one can utilize the fact that the analytical flame is in a steady state and consider that droplets and salt particles are generated reproducibly as if they are in a continuous stream. Identical droplets of the same dimension produced a t the same spot will take exactly the same course of their predecessors. Thus, a t any point in the flame, atoms are produced constantly by a stream of droplets (salt particles). Such a point in the flame can be described as a source of a continuous stream of atoms. The distribution derived from such a source is time-independent. In this approach, the droplet (or particle) chosen actually represents a continuous stream of identical droplets (or particles). For mathematical simplicity, we will take the second approach to describe our system. Consider such a droplet of an initial diameter do introduced into the flame (see Figure 2 ) . The droplet undergoes solvent evaporation on its way up. At a height of z = zo, cm, above the burner surface, evaporation is complete and a spherical salt particle with diameter, d , cm, and density, p , g cm-3, is formed. As the salt particle vaporizes, submicroscopic species, mainly atoms, ions, and molecules of the analyte, are produced and diffuse away transversely. Because of its size (d can be assumed to be negligibly small as compared to do), the salt particle can be envisaged as a point source of analyte atoms. The vaporization is a continuous process as long as the particle lasts. Accordingly, the point source is a moving source. It produces analyte atoms continuously on its way up. This is one of the primary differences between the present model and previously developed models. In the latter cases, salt particles are assumed to vaporize completely in a very short effective distance (3-5). Since the flame is assumed to be a steady flame of uniform temperature and composition, all droplets having the same initial diameter will completely desolvate a t the same height in the flame, while droplets with different initial diameters will desolvate completely a t different levels. In other words, by knowing the droplet size distribution at the burner surface, one knows the height distribution of the point sources. Since the analyte concentration a t the point of observation, (2,R,

.

0

0

0

r

0 '-Ro Hi!Rhl

-

K

Flgure 2. Schematic diagram with dimensions depicting single droplet entering flame and producing atoms

e),from a single point source depends mainly on the position of the source, the knowledge of the height distribution is an essential and indispensable piece of information for the estimation of the overall analyte concentration a t that point. Assuming no mutual interactions between analyte atoms and between salt particles, that is, analyte concentration is additive, then the overall concentration at (2,R, (3) is simply the sum (or more precisely, the integration, if the number of droplets is large) of the statistically weighted concentrations due to individual droplets. Mathematical Analysis of the Stochastic Approach. Consider a droplet (6, 7) of initial diameter do, cm, introduced into a flame. The droplet contains analyte at a concentration ~ . volume of the droplet at z = 0 is of Co, mol ~ m - The


2051

(1)

and so

The mass, mo, in g, of the analyte in the droplet is R

mo = V d o C 0 M ~= - CoMAdo3 (2) 6 where M A is the atomic weight of the analyte in g mol-l. As the droplet proceeds upward through the flame gases at a velocity of u , cm sv1, solvent evaporation occurs (6, 7) resulting in the droplet diameter, d , at any time, t' (t'is taken as the time interval for the droplet to travel from the burner top to any height, z , Le., t' = z / u ) ; the desolvation rate is known (6-10) to be a linear function of the time spent by the droplet in the flame, and so

d 2 = do2 - kdt'

(3)

for the rate of mass decrease with height. Note that both k , and k , are functions of the flame temperature, T. If these functions are known, then the assumption of uniform flame temperature can be omitted. The mass increment, dm, of analyte species undergoing vaporization in a height increment of dz is

If the mass increment is vaporized to form atoms only, the total number of analyte atoms, 6 N ~ ( z , r , 6 )released , within the height increment dz at point (z,r,0)will be

where kd is the rate constant of desolvation, in cm2 s-l. At t' = to, z = 20, the desolvation process is complete. Since do >> d = 0,

and so the flame height, 2 0 , where desolvation is complete, is

where N A is the Avogadro's number, and M A is the atomic weight of the analyte. On the other hand, if submicroscopic species, e.g., ions, molecules, are produced, the fraction of analyte atomized, Pa, i.e., fraction of submicroscopic species in the form of atoms, is less than unity (1, 13-16), then the total number of analyte atomic species, 6Na(z,r,8),released within a height increment, dz, at point (z,r,O) is

Once desolvation is complete, the resulting salt particle will vaporize with a rate constant, k,, in g cm-2 s-l. The rate of solute vaporization, dmldt, is given (11,12) by

N A dm 6Na(z,r,0)= P,GNdz,r,h) = -Pa-dz M A dz

(4)

dm dt

k,d2

--=

where m is the mass of the salt particle at any time, t , thus R

m , pV, = -pd3 6

(7)

The values of m and d depend on the position of the salt particle. p is the density of the salt particle, V , is the volume of the salt particle at any time, t , where t is taken as the time starting from the point of complete desolvation. Since the salt particle is assumed to be spherical,

Note that Pa is a function of height if the flame gas composition and temperatures vary within the height or time interval of interest, but if flame composition and temperature are assumed constant within the height and time interval of interest, then 0, is independent of height. Assuming the particle a t point (z,r,O)is essentially a point source ( I 7, 18)and the number of atomic species of analyte, A , released within the height increment, dz,is 6Na(z,r,0),then the concentration of analyte atoms diffusing to the point of observation, ( Z , R , 8 ) due to the release of 6Na(z,r,6')atoms is given by (see Appendix)

X exp[

and so

where k , is simply the term in brackets to the 2/3-power. Thus, dm --- k sk ,m 2 / 3 dt

According to the definition of t , t = - z - 20 u

u - ~0 [ d ( Z- z

+ + r 2 - 2Rr cos(8 - 0)

) ~R 2

where D is the diffusion coefficient of the atomic species, in cm2 s-l, and all other terms are as described before. As the salt particle moves upward, it vaporizes and releases a different amount of atoms which diffuse to the point (Z,R,8). Since we assume that analyte concentration is ar! additive quantity, the total number of atoms diffusing to the point of observation is the sum of all 6n,'(Z,R,8) released by the salt particle at different positions, i.e.,

where z 2 z o and u is the rise velocity of the particle. Thus

1 dt =-dz

Gn,(Z,R,B) = XGn,'(Z,R,8) or, more precisely,

u

1.7

X exp

2052

0

(16)

[- 2"o [ d ( Z -


z)2

+ R 2 + r 2 - 2Rr cos (e - 0)

vapor, the thermal conductivity of the solvent vapor surrounding the droplet, the molecular weight and density of the solvent, and the transfer number (6,12);the average droplet volume depends upon the design of the chamber and the relative flow rates of nebulizing gas and solution as well as the solution characteristics determining E, and F,. Since each droplet within the area element rdrd0 and the size range do and do 6do contributes 6na(Z,R,8) atoms at

We introduce the step function here when m > 0 m=O

A(m) = 1 =O

to take into consideration the possibility that the salt particle may be completely vaporized before it reaches the level where z = 2. Substituting for 6Na(.z,r,0) in Equation 18 using Equations 14 and 16 gives

6na(z,R,e)= P a N A k c k s

+ + + +

U

m213exp - -[v'(Z - z ) ~ R 2 r 2 - 2Rr cos (0 - 0) - (2 - z ) ] 20 A(m)dz v'(2 - z ) ~ R 2 r 2 - 2Rr cos (e - 0)

1:

~TMADU

1

+

where the integration is taken from the height - 20, where solvent evaporation is complete and solute vaporization begins, toZ, the height of point (Z,R,8),and the terms k,, k,, D, u, and Pa are assumed to be constant with height because of the previous assumption that the flame gas composition and temperature are constant over the height interval of interest. If they are not constants, then they should be included within the integral. If m = mo at z = zo, then it can be shown from Equation 14

(19)

L

point (Z,R,Q),the total atomic contribution at (Z,R,8) is therefore

6nat(Z,R,0) = 6na(2,R,Q)rdrdW(do)6do Because d o is

(23)

to 2 0 by Equation 5,

and so Equation 19 can be rewritten as

6na(z,R,e) =

kcks m01/3 - -(Z - z 0 ) ] 'I3 3u A(m)dz v' (2 - z ) ~ R 2 r 2 - 2Rr cos (e- 0)

Loz

~TMADU

+ +

x exp(-

- U[ ~ ( Z - Z ) ~ + R ~ + ~ ~ - ~ R ~ C O S ( ~ - ~(21) ) - ( Z - Z ) ] ) 20

Up to now we have considered only one droplet and the one particle resulting from it upon solvent evaporation. In analytical flame spectrometry (absorption, emission, or fluorescence), the sample is nebulized into the flame as a droplet mist. If Ns, i.e., dN,/dt, is the total number of droplets introduced per second, then the total number of droplets per second flowing through the burner area rdrd0 at the point (O,r,0)is (NS/~Ro2)rdrd0 of which the total number of droplets having radii between d o and,& 6do is (P(do)6&)rdrd0.The droplet introduction rate, N,, is related to the efficiency of nebulization, en, the sample solution flow rate, F,, into the nebulizerLhamber, the gas rising velocity, u, and the average volume, Vd, of the droplets (i.e., the volume of a droplet multiplied by the total number of droplets gives the total volume of all droplets) and is given by

+

Therefore, Equation 23 becomes 6nat(Z,R,8)= 6na(Z,R,8)p'O d:rdrdzod0 (25) 2 6 where P'(z0) is the droplet distribution transformed to a height distribution. Finally the total number of atoms per unit volume, Ana(Z,R,8) at the point (Z,R,8) due to droplets of all diameters is given by

Ana(Z,R,e) =

1

6nat(Z,R,Q)

(26)

all droplets

Substituting Equation 25 into 26, gives Ana(Z,R,e)

that N , given in Equation 22 is droplet introduction rate in terms of distance rather than time, Le., N,(in s-l) = N,(in cm-l) u . F , is dependent (1,13-15) upon the pressure drop at the nebulizer and the surface tension, density, and viscosity of the solution to be nebulized; e, is dependent (1,13-15) upon the droplet size distribution and the solvent evaporation rate (for a heat-transfer controlled solvent evaporation) which depends upon such factors as the heat capacity of the solvent

. Note

all droplets

Before substituting Equation 21 for 6na(Z,R,8)in Equation 27, it is more convenient to define a new angular parameter,

a, @=0

- 8; and d @= d0

Equation 27 can then be written as

I-

m01/3

kcks ( --

3v X d(2 z ) ~ R 2

2 I3

+ + r 2 - 2Rrcos @ exp

1-

20

+ + r 2 - 2Rr cos @ - (2 - z ) ] } ]

[ d ( Z - z ) ~R 2


(28) 2053

Since the integration of the parameter @ is from 0 to 29, the final form of An,(Z,R,@) after carrying out the integration does not contain a, that is, An,(Z,R,@) = An,(Z,R)

(29)

This is certainly expected for a circularly symmetrical flame.

DISCUSSION The most essential information needed for the evaluation of An,(Z,R) in Equation 28 is the droplet size distribution P(d0)or P’(z0).The functional form for P(do),which has been employed and proved to agree well with observed size distribution for real sprays, is (19-22)

P(d0)= bdoAexp(-adoT)

(30)

where a, b, A, and y are parameters independent of the droplet diameter, do. Equation 30 is the generalized Rosin-Rammler distribution. The special case X = 2 is called NukiyamaTamasawa distribution (21) and X = y - 4 is the RosinRammler distribution (22). The parameters a and b are determined by the total number of droplets per unit volume and by the average droplet diameter, which are defined, respectively, as

Figure 3. Relation between the spatially fixed and the moving coordinate systems

constants, the flame gas composition, and the concentration of analyte a t any point in the flame; thus it can also be estimated stochastically. On the other hand, the efficiency of nebulization, en, is not considered in a fundamental manner, since it can be isolated from the other processes and experimentally measured for any given nebulizer-chamber-burner system. The density, defined as n,l in radiance expressions (15) in atom cm-2, a t a height of Z is n

and

The substitution of Equation 30 into Equations 31 and 32 gives two equations which may be solved for a and b. X and y are parameters which govern the shape of the distribution about the mean. They can be determined by higher order statistical moments, such as the variance, skew or excess, etc. Since only the distribution at the burner surface is needed for the computation of An,, Pido) can be experimentally evaluated with the flame off. in this case, because the desolvation of droplets a t room temperature is much slower than the rising velocity of the spray gases, the distribution will be expected to remain the same within a reasonable height above the burner surface. This gives the experimenter more operational freedom. The rate constants, hd and h,, are temperature-dependent parameters. Although they are treated as constants in this model, their values actually vary with the spacial position of droplets in a non-homogeneous flame. However, unless their functional forms and the temperature profile in the flame are known, treating them as variables only complicates the computation and does not help very much in understanding the basic phenomena in the flame. Meanwhile, we can consider h d and k s as the average values of the spacially dependent rate constants. Their values can be evaluated by taking the average of the results obtained from size measurement of different droplets and salt particles. For large droplets, direct size measurement can be accomplished conveniently by measuring the impressions of the droplets on MgO coated slides (11,23). However, for droplets or particles in the submicron region, accurate size measurement is extremely difficult. More sophisticated methods such as light scattering, x-ray diffraction method, etc., may be needed. Other parameters commonly used in flame spectrometry (see Figure l ) ,such as the fraction of droplets desolvated, ps, and the fraction of particles vaporized into submicroscopic species, &, can also be accounted for stochastically, i.e., by considering each droplet individually. The fraction atomized, pa, can be estimated from the thermodynamic equilibrium 2054

m

n,l = 2 Jo

An,(Z,R)dR

(334

=2

An,(Z,R)dR

(33b)

soL

= An,(Z,R) (2L) (33c) From Equation 28, it is seen that An,(Z,R) decreases gradually toward the edge of the flame. If edge effect is not present, that is, An,(Z,R) is a continuous function and shows no abrupt changes a t the boundary of the flame, An,(Z,R) will become vanishingly small at R 2 L, where L is the radius of the flame at height 2. The upper boundary of the integral in Equation 33a can be reduced to L . The resulting integral is, by definition, the product of the spacially averaged concentration and 2L. Since both An,(Z,R) and L vary as the optical axis moves away from the axis of the circular flame cell, flame profile measurement should be carried out fluorometrically with a very narrow light source such as a laser. The explosion of droplets or particles can be treated stochastically only if the probability of occurring and the probability of fragmentation into various species are known. Transverse migration of droplets and particles may be included in the model, providing the projection of the droplets a t the burner surface can be traced.

APPENDIX Rising Velocity of an Aerosol Droplet or Particle. The transverse velocity of an aerosol droplet of mass, m , diameter, do, and density, p , in a slotted burner flame has been derived by L’vov and co-workers ( 2 ) . The rising velocity, u,, can be derived in a similar manner. The acceleration of the droplet or particle under a driving force, F , can be expressed as du, _ _ -F

dt m where F is the frictional force due to the difference in velocity between the droplet or particle and the flame gases, i.e., , F = 3rgdo(u - u,)

- mg

(A-2)

where mg is the gravitational force, 7 is the average viscosity, and u the rising velocity of the flame gases.On substituting Equation A-2 into A-1, we obtain,


(A-3)

r r 2= R 2

+ r 2 - 2Rr cos (0 - 0)

Substitution of these quantities into Equation B-2 gives Equation 17. because of the fact that m = ( ~ / 6do3p. ) If the particle has already acquired an initial velocity u, = uo at t = 0, the integration of Equation A - 3 gives

ACKNOWLEDGMENT The author thanks J. D. Winefordner for his helpful discussion.

LITERATURE CITED (A-4) Atom Distribution Due to a Single Salt Particle. T o solve for the atom distribution in the flame cell, we employ two sets of cylindrical coordinates. The origin of the spatially fixed reference coordinate system (z,r,B) is located at the center of the burner surface. The other system (d,rr,Or)is located at the center of gravity of the salt particle and is moving without rotation with the particle. The vapor transport equation referred to this moving system is given as the following (18),if a steady state is assumed,

A well known solution of this equation is

which satisfies the boundary conditions n = 0 when z' = *

n=O

r'=w

and

where x 2 = z r 2

+ rr2.

z' and rr are related to z,r, and 0 by (Figure 3 ) z'=Z-z

and

(1) C. Th. J. Alkemade, "Fundamental Aspects of Decomposition. Atomization, and Excitation of the Sample in the Flame," Chapter in "Flame Emission and Atomic Absorption Spectrometry", Vol. I, F. A. Dean and T. C. Rains, Ed., Marcel Dekker, New York, 1969. (2) B. V. L'vov, L. P. Kruglikova, L. K. Polzik, and D. A. Katsdov, J. Anal. Chern. USSR,30, 545 (1975). (3) B. V. L'vov, L. P. Kruglikova, L. K. Poizik, and D. A. Katskov, J. Anal. Chem. USSR,30,551 (1975). (4) H. A. Wilson, Phil. Mag., 24,(6), 118(1912). (5) H. Bavinck, Rept. TW 98, Mathematical Center, Amsterdam, 1965. (6) J. M. Beer and N. A. Cahigaer, "Combustion Aerodynamics", Applied Science Publishers, Ltd,, London, 1972. (7) F. A. Williams, "Combustion Theory", Addison Wesley, Reading, Mass., 1965. (8) N. C.Clampitt and G. M. Hieftje, Anal. Chern., 44, 1211 (1972). (9) F. A. Williams, in "8th Symposium on Combustion", Williams and Wilkins, Baltimore, Md., 1962, p 50. (IO) D. B. Spalding, in "4th Symposium on Combustion", Williams and Wilkins, Baltimore, Md., 1953, p 847. (11) G. J. Bastiaans and G. M. Hieftje, Anal. Chem., 46, 901 (1974). (12) P. W. Jacobs and A. Russel-Jones, J. fhys. Chern., 72, 202 (1968). (13) J. D. Winefordner, R. G. Schulman, and T. C. O'Haver, "Luminescence Spectrometry in Analytical Chemistry", John Wiley, New York, 1972. (14) C. Th. J. Alkemade and P. J. T. Zeegers, "Excitation and De-excitation of Atoms," Chapter in "Spectrochemical Methods of Analysis", J. D. Winefordner, Ed., John Wiley, New York, 1971. (15) J. 0. Winefordner, V. Svoboda, and L. J. Cline, Crit. Rev. Anal. Chem., 1, 233 (1970). (16) P. J. T. Zeegers, R. Smith, and J. D. Winefordner, Anal. Chem., 40 (13), 26A (1970). (17) W. Snelleman, Ph.D. Thesis, University of Utrecht, Utrecht, The Netherlands, 1965. (18) P. W. J. M. Boumans, "Theory of Spectrochemical Excitation", Adam Hilger, Ltd., London, 1960. (19) Y. Tamasawa, Tech. Rept. TohokuUniv., 18, 195 (1954). (20) Y. Tamasawa and T. Tesima, Bull. JSME, 1, 36 (1958). (21) S. Nukiyama and Y. Tamasawa, Trans. SOC.Mech. Eng., Tokyo, 5, 62 (1939). (22) P. Rosin and E. Rammler, Z.Ver. Deutsch. lng., 71, 1 (1927). (23) G. M. Hieftje and H. V. Malmstadt, Anal. Chem., 40, 1860 (1968).

RECEIVEDfor review April 21, J976. Accepted August 23, 1976. Presented in part at the ACS Award in Analytical Chemistry Symposium, 171st National Meeting, American Chemical Society, New York, N.Y., April 4-9, 1976.


2055

Theoretical treatment of a model for analytical flame spectrometry

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