A theoretical analysis of the diffusion process in flameless atomizers

SpecfrochimlcaActa, Vol. 33B, pp. 695 to 700 Pergamon Press Ltd. 1978. Printed m Great Britain

A theoretical analysis of the diffusion process in flameless atomizers* H. FALK Zentralinstitut fiir Optik und Spektroskopie, AdW DDR, 1199 Berlin, Rudower Chaussee 5, German Democratic Republic (Received

3 April

1978)

Abstract-The density distribution of the atoms and its variation with time are calculated for the filament and the tube atomizers. A theoretical model is used, which is valid for fast heating of the furnace and for predominant diffusion. A linear approximation of the density distribution is applied in the model of the vapour cloud. On this basis the dependence of the absorbance on time is calculated.

1.

INTRODUCTION

ANALYTICAL methods using atomic spectroscopy atomization of the substance under investigation is necessary. The density and lifetime of the vapour cloud of the sample determine the efficiency of atomic emission, absorption or fluorescence. Therefore the processes in the atomizer are of considerable interest. Especially it would be important to estimate the upper limits of the density and lifetime of the vapour cloud. For the tube furnace these limits have been estimated by L’vov [l]. In his approach the initial contribution to the absorption signal from the vapour cloud that propagates from the centre to the end of the tube is disregarded. The validity of this approximation is considered below. In addition to the flameless atomization in a tube furnace the open atomizers, for instance, the carbon rod or the filament, are of practical interest for atomic absorption. Moreover these atomizers are of particular importance for atomic fluorescence. In this case a high density of the sample atoms is needed since the saturation intensity with the exciting laser [2] can be reached only in a small volume. The lifetime of the vapour cloud was determined experimentally by several authors [3-51. In [S] the time constant of the evaporation process of the sample was estimated for the filament atomizer. The influence of diffusion upon the lifetime was not observable since the sample cloud was transported by the shield gas flow. A satisfactory agreement of the calculated and measured evaporation rates was found for elements with high evaporation temperature. In order to obtain a high concentration of analyte atoms and a long residence time it is suitable to use the stop-flow regime. In this case the propagation velocity of the vapour cloud is determined by diffusion. For the filament atomizer this process is more or less modified by convection effects. One obtains a high density if the atomization time is short in comparison with the diffusion time constant [4]. In order to calculate the time constant of diffusion for the filament and tube atomizers we make the following assumptions :

FOR

(i) The original sample is concentrated in a volume that is small in comparison with the extent of the vapour cloud. (ii) zA 6 zD where TA= atomization time and 5/l = half-life period of the vapour cloud. (iii) The mean free path of the analyte is small in comparison with the diameter of the cloud. (iv) The influence of convection is negligible. * Dedicated I]

21 31 .4] .5]

to the memory

of Professor

HEINRICH

KAISER

B. L’VOV, Atomic Absorption Spectrd Andysis, p. 201. Hilger, London (1970). S. NEUMANN and M. KRIESE, Spectrochim. Actn 29B, 127 (1974). R. E. STURGEON, C. L. CHAKaABARn, I. S. MAINES and P. C. BERTELS, Anal. Chem. 47, 1240 (1975) R. E. STURGEON, C. L. CHAKRABARTI and P. C. BERTELS, And. Chem. 47, 1250 (1975). D. J. JOHNSON, B. L. SHARP, T. S. WEST and R. M. DAGNALL, Anal. Chrm. 47, 1234 (1975).

695

696

H.

FALK

Fig. 1. Cross section of the vapour cloud determined by diffusion: r = spatial coordinate; rR(t) = distance of the vapour border from the centre 0; 0 = sample position: (I = distance of the ray for the absorbance measurement from the filament surface.

2. FILAMENT ATOMIZER

With depends notation

assumptions only upon is used :

(i)-(iv) the density in the vapour cloud above a flat atomizer the distance from the sample position (see Fig. 1). The following

n(r, t) = number density in the vapour cloud (m- 3); = number of sample particles (no unit); No V(t) = volume of the vapour cloud (m3); r,(t) = radius of the vapour cloud (m); t = time(s). If atomization

begins at t = 0, we have fl(r,O)=noo

for

O~r~ro

(14

r > r.

(lb)

where r. = rx(0), and n(r,O) = 0

for

Also, 23K V(0) = V. = - vi; No = Vonoo 3

For simplification

we assume that V(tl) = 21/,;

rR(tl) = 2113ro

(2)

and for t 2 tl that @, t) = if no(t) = ~(0, t). The hemisphere

in Fig. 1 contains

the particle

*iI

n(r’, t)2zr”

s The concentration

gradient

dr’=

-

(3)

r/h(t)]

number z no(t)ri(t)

= No.

(4)

0

is a@, _~-dr

Therefore

n0W[l

t)

n0W

(5)

r&)'

we have a radial mass transport dN,(t)

where D(m*s-‘)

= -D27cri(t)

is the diffusion

constant.

an(r, 0 ___ar dt = 2nDno(t)rR(t)

dt,

(6)

A theoretical

analysis

of the diffusion

process

in flameless

atomizers

697

Using equation (4) for the growing hemisphere we find dN,(t) = ; [n& + dt)r:(t + dt) - no@ + dt)r;(t)]

(7)

With the aid of equations (4), (6) and (7) we obtain

ano(t + dr)ri(t)

= No -

2nDno(t)rR(t)dt

(8)

and after some elementary calculations

no@ + dt)

- no(t)

dt

dno(t) =p= dt

- B@(t)

(9)

where B = 322’3D (10)

Z/3 2 HO r0

The solution of equation (9) is -312

for t 2 tl. From equation (4) we have no(t + dt)ri(t + dt) = no(t)ri(t)

(12)

and neglecting all terms higher than first order in dt we obtain r

(t)

R

dno(t)

dt

drdt) =

+ 3no(t) T

o

.

(13)

With condition (2) the solution of equation (13) is rR(t) = 2”3[2D(t - t1) + rQ1’2

(14)

for t 2 tl. If at time t2 the density in the centre is 4 of the initial value, then from equation (11) it follows that t2 = tl + 0.294;.

(15)

The radius of the hemisphere at t2 is rR(t2) = 22’3r0.

(16)

With the plausible assumption t2 - tl x tl the “lifetime” of the vapour cloud is 71

z

2(t, - tl) = 1,92. 1019m”2d2p1/3T- 5’6N$‘3

(17)

if the diffusion constant [6] is expressed as D=0,21(kT)3’2p-1d-2m-1’2

(18)

and the validity of the ideal gas law is assumed. Equation (18) holds for a gas mixture consisting of two components with d=dr

+d, 2

m=---

2mm2

ml

(19)

+m2

where dl and d2 are the diameters of the particles (m), ml and m2 their masses (kg) of the particles, p is the gas pressure (Nme2), 7 the gas temperature (K) and k the Boltzmann [6]

H.

EBERT,

Physikalisches

Taschenbuch,

p. 219. Vieweg, Braunschweig

(1967).

H. FALK

698

Fig. 2. Normalized

theoretical

density (A) in the centre and absorbance the filament as functions of time.

(B) along

the surface

constant. If we measure the absorption parallel to the surface of the atomizer u(m) and a frequency v we find for the absorbance (see Fig. 1) (ri - u2)1’2 -Tln[/$-

A,. = 0.43430&

l11’2+!L]i

of

at a distance

(20)

i where or, = absorption cross section (m’). Directly absorbance during the time interval 0 5 t i z1 is A, % 0.2870,,nooro If we apply the integration

method

-

above

the surface at a = 0 the mean

NA’3.

(21)

for the measurement,

then the signal is

At, - No.

(22)

Figure 2 shows the dependence of the density at r = 0 and of the absorbance at a = 0 on time as dictated by equations (11) and (20). The absorbance decreases by a factor I/e during time TV. To illustrate the order of magnitude of the time constant as determined by diffusion we consider the atomization of 5Opg NaCl in an argon atmosphere at a pressure of 1 bar and a temperature of 2800 K. From equation (17) we find r1 = 40ms. The above mentioned description of the development of the vapour cloud of the open atomizator is a borderline case. Under real conditions the convection modifies the diffusion process but it is impossible to stabilize the vapour cloud in the open system longer than given by the time constant of the diffusion.

3. TUBE FURNACE We consider a cylindrical tube with a constant temperature along the tube. Sample losses through the walls are disregarded. We assume a uniform vapour density across the cross-sectional plane of area S. If the sample is in the centre plane of the tube, the vapour density is symmetrical relative to this locus. Analogously to the previous section we assume a linear decrease of the density from the centre to the end of the tube (see Fig. 3). At the moment of evaporation at t = 0 we have n(xo,O) = 0 and for x,(t) 5 I1 we assume ll(.Y[, t) = no(t)

1- ; (

Then it follows that

s XI

S

n(x’, t)ds’ = $.

0

I>

(23)

A theoretical

analysis

of the diffusion

A

process

in flameless

atomizers

699

n (x-t )

n( t ln(t+dt)‘

I I I I x,(t)

(1 x,(;+dt

Fig. 3. Diagram

of the density

distribution

in the tube particles.

atomizer.

x

)

n(.u,t) = number

density

of

The density in the centre is no(t) =

No1 s x,(t)

(25)

and the number of the particles transported by diffusion is

dN = ‘$ dnO

(27)

and equation (25) we have dno(t) = 2Dno(t) =

No dxAt) s

dt

The solution of equation (28) is xl(t) =

20 r/2

(29)

The time needed for the expansion of the vapour from x = x0 to x = 1r is z2

=

4&: - 11x0)

(30)

and for l1 $ x0, which holds in the case of small samples we have

l2

52

=

4&

(31)

In the time interval 0 5 t 5 z2 the absorbance is Av2 = 0.43430, :.

(32)

700

H. FALK

Fig. 4. Normalized

absorbance

For t > z2 the number of sample therefore the absorbance too :

as a function

particles

of time for the tube atomizer

in the tube decreases

exponentially

g,No AP3 = 0.4343 __ S

[l]

and

(33)

where l2 73 =

Figure 4 shows the time dependence is suited for measurement [4], becomes

s

of the absorbance.

Av2~2 = 0,25

and

5

(34)

*&

m s0

over A”(t), which

0.4343 o,.Nol: (35)

SD

Ayj(t’)dt’ = 0,32 0.4343 o,,Nol; SD

0

or

The integral

0.4343 o,Nolt sD A,.s(t’)dt’ = 0,50 -

(36)

(37)

Comparison of equations (35), (36) and (37) respectively shows that they contribute comparable portions to the signal. Therefore we cannot neglect the atomization phase of the propagation of the vapour cloud from the centre to the ends of the cuvette.

4. CONCLUSION The above mentioned calculations permit a comparison of various types of flameless atomizers or their adaption to the measurements respectively. For the tube atomizer it was experimentally shown, that the diffusion process prevails [l]. In contrast to that we have not yet enough experimental data to decide this problem for the open atomizers too. In dependence on the experimental results we may have to complete the diffusion theory by considering the convection influence. Corresponding measurements for the investigation of this question are in preparation.