Demixing effect induced by the radial electric field in the cylindrically shaped d.c. plasma. Modeling and experimental investigation

SPECTROCHIMICA ACTA PART B

Spectrochimica Acta Part B 53 (1998) 81-94

ELSEVIER

Demixing effect induced by the radial electric field in the cylindrically shaped d.c. plasma. Modeling and experimental investigation 1 M i r j a n a S. P a v l o v i d , M o m i r M a r i n k o v i d Laborato~' of Physical Chemist~', The Vin~'aInstitute, PO. Box 522, 11001 Belgrade, Yugoslavia

Received 17 June 1997; accepted 17 October 1997

Abstract

Sample introduction and a demixing effect in the plasma of the d.c. arc in argon with aerosol supply are studied using spectroscopic techniques. The observed deviations from the uniform distribution of the partially ionized plasma gas components are correlated to the radial electric field in the arc column plasma. A mathematical model is proposed for the quantitative treatment of the radial field effect. The radial electric field is assessed from the radial distribution of the electron pressure. The proposed model is used in two ways: (a) starting from the close-to-real arc temperature distribution, other parameters are calculated assuming that plasma is in local thermodynamic equilibrium. The calculations are performed for mixtures of two gases: majority gas with high ionization energy and minority gas with low ionization energy at low concentration. The results obtained show that the partial pressure of an easily ionizable component decreases from the periphery towards the arc column axis. The size of the decrease ranges from a negligible value up to four orders of magnitude, depending on the ionization energy of the minority gas. (b) For the experimental verification of the proposed model, all the variables governing the sample entry into the plasma, including the initial partial pressure of the minority gas, are determined by independent spectroscopic techniques. Vapors of Ba, Ca, Fe and Zn are utilized as a minority gas. These values are then used to calculate the radial distribution of the minority gas partial pressure according to the proposed model. Good agreement between the experimental and the calculated values is observed. © 1998 Elsevier Science B.V. Keywords: Demixing effect; D.C. plasma; Radial electric field; Emission spectrochemical analysis; Argon

1. I n t r o d u c t i o n

The effect of mass separation of various species in the arc plasma was noted many years ago [1]. Its theoretical treatment is detailed in Refs [2] and [3]. Significant work has been done recently on the demixing effect in thermal plasmas by a number of workers [4-8]. A more comprehensive list of references is given by Murphy [9]. Spectrochemical excitation sources were considered in the papers published by This work was supported by the Science Foundation of Serbia, Yugoslavia.

Vukanovi6 [10] and Shirrmeister [11,12]. To our knowledge, since that time that approach has never been used again in the spectrochemical literature. With the advent o f d.c. plasma sources with aerosol supply, difficulties with aerosol sample introduction into the arc plasma have been encountered. They have been ascribed to aerosol particle rejection by the plasma due to a high temperature gradient [13-15]. However, this approach has only offered a qualitative interpretation. Recently, we have recognized the importance o f the effect of radial electric field and attempted to model and study it experimentally [ 16,17]. In a cylindrically

0584-8547/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved PII S0584-8547(97)00116-X

82

M.S. Pavlovi{, M. Marinkovi{/Spectrochimica Acta Part B 53 (1998) 81-94

shaped d.c. arc discharge plasma, due to gradients of charged particles (and temperature) in a radial direction, electron and ions migrate towards the arc periphery. Because of the higher mobility of electrons, a rising radial electric field retards the electrons and accelerates the ions, ensuring the same drift velocity of electrons and heavier ions. In steady state, an equilibrium among partial pressure gradients, friction forces that arise due to the relative motion of the individual constituents, and the radial electric field is established. If the gas mixture components are ionized to a different degree, owing to their different ionization energies, the radial electric field will produce partial demixing of the initially homogeneous mixture. This is manifested as a partial pressure depression of the gas characterized by a lower ionization energy in the high temperature plasma zone. In the first part of this paper a mathematical model is proposed, and the radial distributions of plasma components are calculated. The local thermodynamic equilibrium (LTE) condition is assumed to be satisfied. The second part of the paper describes an experimental study of the effect on the low current arc in argon that generates a plasma with high temperature and electron concentration gradients [ 17], which favor emergence of the effect. To obtain a more reliable comparison of the experimental data with those predicted by the model, the calculations are carried out with all relevant plasma parameters that were determined experimentally.

2. Development of the mathematical model A d.c. arc plasma at atmospheric pressure was used as an object for modeling the effect of the radial electric field on the mass separation of the plasma components. For the arc plasma, the following assumptions were adopted: 1. The arc burns in a static gas at a pressure of 1.01325.105 Pa (1 atm). 2. The arc column is infinitely long so that influence of electrode regions could be neglected. 3. The arc column is cylindrically shaped and confined in a tube, 16 mm in diameter, with cool walls. 4. The arc plasma is in local thermodynamic equilibrium (LTE).

5. For the sake of simplicity, it is assumed that the arc discharge is sustained in a mixture of two g a s e s - - a majority gas with a high ionization energy and a minority or trace gas with a low or medium ionization energy at low concentration. The vapor of a chemical element could be considered as a trace gas. In spectrochemical excitation sources, the element sought or analyte (the latter term is recommended by IUPAC [18]) exist as a minority or trace gas due to sample dilution by plasma gas. Further, in this paper, the fill gas will refer to the majority gas, and the analyte to the trace or minority gas. 6. Plasma temperature is not influenced by small contents of an analyte. The fill gas and analyte were hypothetical elements with anticipated properties as follows: 7. Energy of ionization of the fill gas is 14 eV, and the energy of ionization of the analyte ranges from 4 to 10 eV. 8. The ratio of the ion partition function to the atom partition function is equal to unity and is not dependent on temperature. 9. The volatility of the analyte or its compound is high so that no solid or liquid particles are present in the plasma. 10. The analyte does not form stable compounds, even in zones of the lowest temperature. Because of the high pressure, 1.01325.105 Pa, it seemed appropriate, to a first approximation, to use a single temperature for describing the system. Other variables of the system were partial pressures, p, and drift velocities, v, of the constituents. Subscripts are used to denote the constituent (fstands for fill gas, a for analyte, and e for electrons) and superscripts denote the charge of the particle (0 stands for neutral, and 1 for singly charged). The partial pressure of the constituent is related to number density, n, by p=nkT,

(1)

where k is the Boltzmann constant and Tthe temperature. Low ionization of the fill gas, and the supposed small fraction of analytes, together with some of the already listed restricting assumptions (4, 6, 9 and 10), allow important simplification in treatment of the system yielding a remarkably compact expression for the

M . X PavloviO, M M a r i n k o v i O / S p e c t r o c h i m i c a

analyte radial distribution. This approach is particularly justified for plasma regions beyond 2.5 mm distance from the axis, where the prevailing percentage of demixing for analytes of low and medium ionization energies occurs. It should be noted that this region is significant for spectrochemical excitation. For further treatment of the system, the following assumptions were adopted: (a) electric force is the only external force; (b) thermodiffusion, viscosity, and chemical reactions are neglected; and (c) the plasma is quasi-neutral. It was found that, for the approximate calculation of the analyte radial distribution, the following relations were sufficient [ 19]: (a) Saha's equation for an analyte ionization equilibrium

P~e

equation [3,20]: ,.0). (o) \ 1/2 ~(10)_ 3t300) ' 2.k.V '"a m). | sqt" - 8~qf 7r m(1) +m(°)J

(2)

P, where U~°) and U~1) are the partition functions for analyte neutrals and ions, respectively, and E~ is the ionization energy of the analyte atoms. (b) An ion can be formed only when the atom disappears simultaneously at the same point. That means that the sum of the corresponding particle streams has no source; consequently the equation of continuity resumes the form: (3a)

V "k~ta l.(0) "V .-I(,0)_. (1) "V .-z(, a ~- gl O a 1)~) = 0 '

written for analyte ions and fill gas neutrals. Because of the small number density of particles other than fill gas neutrals (fill gas ions, analyte neutrals and ions and electrons), the corresponding terms for the friction forces that involve the product of two small number densities, were omitted from Eq. (4a), Eq. (4b), Eq. (4c). (d) Expression for the radial electric field, which is applicable when the condition for quasi-neutrality is satisfied [ 12,20]:

E= - kT.V(lnp~).

or in cylindrical coordinates (1) / (l)x

, "~v, j~ +no "tvo )~ =0.

(3b)

(c) Hydrodynamic equations of motion for fill gas and analyte constituents: 0 = - Vp~°)

(4a)

0 = - VP(°) - n(°)'n(°)'¢(°°)o y "af ",~ozdO)--V~ 0))

(4b)

t'a

""f

"~;af "~Va

~o)~ . . . .

-- v f ] - - . a

(1).~

..a

(6)

Although a more complete set of equations could be used for the description of the plasma studied, it was found that the above approach was satisfactory for the purposes of this work. In addition, it emphasizes the influence of the radial electric field on the analyte distribution. The equations are combined as follows. Firstly, Eq. (4b) and Eq. (4c) are summed, and taking into account that P a = l J_(0) a _ T I J~(1) a , it is found that: (1).~7+..(o) ..(0) ~(o0) ~o) ~0)~ 0=-Vpa+Za'n a n a "nf "gay "~, a - f J -4-..(1) ..(0) ~.(10) ¢.--~(,l)

r, a ",t/. "¢~af "~'Va - f

0 = -Vp(. 1)--(1) .(o) ~(10) t~l)

(5)

e

Uad) f Ei "~ =6.67.10 2.T(5/2). u~o).exp ~ ~--f) ,

h(O) [.(0)'~ _

83

Acta P a r t B 53 (1998) 81 94

~.

(4c)

where if7 is the radial electric field, Za the electric charge of the analyte ion, and ~ the coefficient of friction that is connected to masses, m, and gas kinetic cross sections, Q, of colliding particles by the

V-~0)

)"

(7)

At this stage, a further approximation was made by assuming that the coefficients of friction of the analyte ~(0o) = ~ o ) = ~. atoms and ions are equal, that is say Because the ion and atom masses are almost equal, the term in parenthesis in Eq. (5) is identical for both cases, while the gas kinetic cross-sections were presumed to be similar as well. For the arc plasma modeled in this work, the degreeof ionization of the fill gas at the arc axis was small (6.10 3). It was much smaller in the off-axis regions, and beyond 2.8 mm distance, the calculated ionization fraction of fill gas was < l0 -5. The initial analyte number density fractions considered, na/(n/+ ha), were also small, commonly < l0 -4. In the first approximation, all these small number densities could be disregarded, thus reducing the equation of motion for fill gas to Eq. (4a). As a

84

M.S. PavloviO, M. MarinkoviO/Spectrochimica Acta Part B 53 (1998) 81-94

consequence, we have v(°)= 0 and then we obtain: f 0 = - V p a -~~ .,,0).~_ ,,(0).e 4,,(0).,~,o) ~_,,0).,~, J)~ --~a"a

~

"f

'~ ~,"a

Va

--"a

Va

f'

(8)

After substitution of the expression for if? from Eq. (6) into Eq. (8), and taking into account Eq. (1) and Eq. (3a), according to which the expression in parentheses should be equal to zero, we finally obtain: V lnpa = - X'V In Pe

(9)

or, in cylindrical coordinates 0(ln p~) _ Or

O(lnpe ) X - - , Or

(10)

where X is the fraction of the ionized analyte, X=tl(al)l(n(aO)+tl a(I)-, ) : t l a(1),/ n a. It can be seen that the radial distribution of the analyte partial pressure, po, depends on the fraction of the ionized analyte, X, and on the slope of the In Pe curve. In the absence ofanalyte ionization, the analyte pressure will be constant across the arc column, otherwise it will show a depression at the axis.

3. Modeling of the analyte distribution Modeling of the radial distributions of the analyte pressure and other plasma parameters that depend on analyte concentration has been performed by numerical solution of Eq. (10), assuming that the conditions for LTE are satisfied. Because of the inter-relation between the distribution of the electron pressure and the analyte pressure, an iterative method was used. The input parameters were the energies of ionization of the fill gas and of the analyte, and the initial number density fraction of the analyte. The radial distributions of the following plasma parameters were calculated: electron pressure, fraction of the ionized analyte, radial electric field and analyte partial pressure. Two series of data were obtained: (a) the initial number density fraction of analyte was kept constant, while the analyte ionization energy was varied; and (b) the analyte ionization energy was kept constant, while its concentration was varied.

3.1. The radial temperature profile It was assumed that a small concentration of the analyte will not change the temperature of the arc

plasma. It was presumed that the temperature profile, in the range 0-7.5 mm, could be described by the relation:

1 r]~' T=T° ( 1 + wJ

(11)

where r is the radius, and To and w are constants (To = 8788 K, w = 3.538 mm). The constants To and w were so chosen to fit the experimental temperatures obtained with the arc plasma described previously [17]. The temperature above 7.5 mm does not follow Eq. (11) and is expected to drop sharply down to the temperature of the cool walls surrounding the arc column. This region was not included in our modeling since it does not affect the analyte distribution because of the negligible analyte ionization in it.

3.2. The electron pressure After the composition of the gas has been chosen, the plasma parameters were calculated assuming the validity of LTE. The electron pressure distributions were calculated for mixtures of fill gas and various analytes in different proportions. In the first series of calculation, the ionization energy of the fill gas was 14 eV and the ionization energy of the analytes varied from 4 to 10 eV. From the data graphically presented in Fig. 1, the following conclusions can be derived. 1. The electron pressure is appreciably affected by analyte when its partial pressure is comparable or higher than the electron pressure produced by the ionization of main gas alone. In the particular case, shown in Fig. 1, the initial analyte (not ionized) partial pressure was 0.1 Pa. This pressure was comparable with the electron pressure that could have been created by the ionization of the main gas alone at a radius of 3.5 mm. For the regions closer to the arc axis, the electrons resulting from the ionization of the main gas prevailed, and the analyte contribution is not noticeable on the graph. 2. Analytes with low ionization energy (4-6 eV) produce the horizontal portion on the distribution curve; the length of this portion is larger when the ionization energy of the analyte atoms is lower. In this region, the electrons are mainly contributed to by the ionization of the analyte, and the analyte was completely ionized.

M.S. Pavlovid, M. Marinkovi(/Spectrochimica Acta Part B 53 (1998) 81-94

85

4o

!

1o

]

2.0

!

0.0

-~

-2.0

4

~

i \ 0.5

\/// /

10

O -4.0

-~.°~

',\ \ \

\7 \

14__

-8.0 i

0

........................... 1 2 3

4

5

;' ~ \ " \ 6

7

' ....

...... Ti ......... ~........ ;~ ....... '4........ ; ........ 6 ~ 7

Radius, m m

Radius, m m Fig. 1. Radial distribution of the electron pressure calculated for various mixtures of fill gas (E~ = 14 eV) and an analyte. Analyte ionization energy varied from 4 to 10 eV. Initial number density fraction was 10-6 for all analytes. The number on each curve stands for analyte ionization energy.

3. The electron pressure corresponding to the horizontal part is only half the value that would be obtained if the analyte were fully ionized and evenly distributed across the arc column, that is, if analyte demixing were absent. In case shown in Fig. 1, an initial analyte number density fraction of 10 -6 would produce the horizontal part of the electron pressure curve at 0.1 Pa, but the calculated value was 0.05 Pa.

3.3. Ionization of analyte

........

Fig. 2. Radial distribution of the analyte ionization degree calculated for various mixtures of fill gas (E~ = 14 eV) and an analyte. Analyte ionization energy varied from 4 to 10 eV. Initial number density fraction was 10 ~ for all analytes. The number on each curve stands for analyte ionization energy.

>"

15!I

10 9

-U 1o

The fraction of ionized analyte was calculated under the same assumption as for the electron pressure made in the previous section. The curves in Fig. 2 differ slightly from those obtained by assuming an even analyte distribution.

| o.oJ~ 0.

3.4. The radial electric field For various combinations of two gases, fill gas and analytes of different ionization energies, the radial electric field distributions were calculated and

. . 1. . . 2

3

4

5

Y

3".......

Radius, mm Fig. 3. Radial electric field calculated for mixtures of fill gas (E~ = 14 eV) and an analyte. Analyte ionization energy varied from 4 to 10 eV. Initial number density fraction was 10-6 for all analytes. The number on each curve represents analyte ionization energy.

86

M.S. Pavlovik, M. Marinkovik/Spectrochimica Acta Part B 53 (1998) 81-94

shown in Fig. 3. Eq. (6) is only applicable when the concentrations o f the charged particles are so large that a sufficiently strong electric field can be created by a small displacement of charge at any point so that assumption n~o, ~- n~ would still be valid. The limit can be estimated by the Gauss law: eoV'/~ = eSpe

0.0~~ -1.0!

/

//

76

5

'4

(12)

kT

and in the case o f cylindrical symmetry 1 tg(r E~) eo

r

-

-

-

Or

~'~ -3.0

e6p~ - -

kT '

(13)

where eo is the permittivity o f the vacuum and dpe = P~o, - P c , is due to the disturbance of the charge balance. The electric field and the other parameters were calculated only when dpdp~ was smaller than 0.01 and curves are drawn to that point. The radial electric field is not appreciably influenced by the analyte in the range from the arc axis up to 3 mm distance. Above 3 mm, the presence of the analyte atoms in the plasma decreases the field, depending on the analyte ionization energy. The effect is most significant for the analyte with an ionization energy o f 4 eV (Fig. 3, curve 4). The sharp fall of the radial field is followed by a range where the field strength is almost zero. Further, it rises again, and beyond 6.5 m m distance, it starts to decrease slowly. 3.5. Analyte distribution

Analyte partial pressure monotonously decreases from the periphery of the arc column toward the axis, the decrease being the largest at the axis (Fig. 4). For the easily ionizable analytes, horizontal steps at the - 0.3 level on log scale, relative to the initial value at the arc periphery, are discernible on the distribution curves. These horizontal steps extend over the range where the most of the electrons originate from the analyte atoms, and the analyte atoms are completely ionized. Closer to the arc axis, depletions of the analyte pressure are considerably increased for the analytes with low ionization energy and may reach more than four orders o f magnitude. From the second series of calculated plasma parameters, where the analyte concentration was varied, only a graph representing the radial distribution of the

-4.0 ~4,5 -5,0-~,

i i i H,

0

i r i ii~r~l

1

i i FIn

i .....

2

r i Hir

IIl,ljlllrllnllll

3

4

H

5

irl

H

n I .....

6

i1,11

.....

7

i~rl

~

8

Radius, mm Fig. 4. Radial distribution of the analyte partial pressure (p{a°}+p~l}) calculated for various mixtures of fill gas (E, = 14 eV) and an analyte. Analyte ionization energy varied from 4 to 10eV; P~,8 was the initial partial pressure of the analyte. The number on each curve represents analyte ionization energy.

1oI 0.0 -1.0;

/

-2.0 l~q

/

30

1

/

4 ° . 11

/

-5.02 1~ -6.0

1,

-7.0

1,

-8.0 1 -9.0 0

.........

I .........

1

I .........

2

n

3

........

r .........

4

i .........

5

i .........

6

i .........

7

8

Radius, mm Fig. 5. Radial distribution of the analyte partial pressure calculated for various mixtures of fill gas (E, = 14 eV) and the analyte (E, = 5 eV). Initial numberdensity fractionofanalyte varied from 10 ~0to 104 . The number on each curve represents the initial number density fraction of the analyte.

M.S. Pavlovik, M. Marinkovff'/Spectrochimica Acta Part B 53 (1998) 81 94

40i - Log p~ 2.0

~

0.0

e-~ 2O

° M

t - 4 . o ~

-6.0

0

Log p~

, .........

~

1

2

]

[ ........

3

~ll

,, ii

i i [ .........

5

[ .........

6

~

7

8

87

following steps: (a) experimental determination of three variables by independent, mainly spectroscopic, methods; (b) calculation of the distribution of one variable according to Eq. (10) by using experimental data of the other two variables; and (c) comparison of calculated and experimentally obtained data. Concerning the proposed procedure to verify the model, it should be noted that the electron number density in the arc plasma region, where the analyte concentration undergoes the greatest change, is a few orders of magnitudes lower than the density required for attainment of LTE [21]. Indeed, an experimental study of a similar arc showed that an appreciable deviation from LTE conditions exists [22]. Therefore, an eflbrt was made, whenever possible, to use measurement techniques that are either independent or only weakly dependent on the LTE assumption.

Radius, mm Fig. 6. The analyte partial pressure and the electron pressure curves calculated for an hypothetical analyte having an ionization energy close to zero. The ionization energy of the fill gas was 14 eV, and the initial number density fraction of the analyte was 10 6. The curve pattern is symmetrical with respect to the horizontal line at -1.

analyte partial pressure is given in Fig. 5. For an analyte ionization energy of 5 eV, it is seen that the partial pressure depression is strongly dependent on the analyte concentration. In this case, it ranges from two and a half orders of magnitude (for an initial number density fraction of 10 -4) to over eight and a half orders of magnitude (for an initial number density fraction of 10-~°). It follows from Eq. (10) that the greatest possible depression of the analyte pressure, when the analyte is completely ionized, is limited by the electron pressure distribution (Fig. 6). It is interesting to note that the horizontal parts on the electron pressure, electric field, and analyte pressure curves (Figs. 1,3 and 4) are obviously correlated, indicating a common cause for their behavior.

4. Experimental testing of the model The fact that all three variables of Eq. (10) (O(lnpo)/ Or, X, 0(In p J/Or) can be evaluated experimentally makes the validation of the proposed model relatively easy. Testing of the proposed model consisted of the

4.1. Stabilized arc device

Experimental measurements were carried out on the plasma of a 7.5 amp stabilized d.c. arc in argon with an aerosol supply. The arc device, the instrumental system (nebulizer, monochromator, and measurement electronics) and the experimental conditions have been described elsewhere [17]. An analytically usable portion of the arc column was 50 mm long, cylindrically shaped, and suitable for observation in the axial direction, that is, for "end-on" observation. Effective elimination of the plasma plumes from the emission, and, particularly, from the absorption path made possible the evaluation of the spatially resolved local plasma parameters without the use of Abel inversion. 4.2. Temperature measurements

Temperature measurement was described and discussed elsewhere [16,17]. In this work the measurements were extended towards the arc column periphery by using the absorption technique, since the spectral emission of thermometric species was too low to be measured. The weighted experimental data, fitted by rational function, are shown in Fig. 7. 4.3. Electron number densi~ measurements

Owing to the large span of values involved, electron

88

i0o_

bd 8000I

M.S. Pavlovik, M. Marinkovik/Spectrochimica Acta Part B 53 (1998) 81-94

3.0

2o o

~

1.0

6000

~,~ o.o Z

~N~

l:a., ,000

~'d -2.01 -3.0

2000 -4.0

0

iillinllllrlll

0

n IIFIIIIHIII~IIIIII,I

1

2

3

,ll

4

M Illlllll

n IIIIH

5

rll ,lit

6

iiiFr,i,

7

-5,0

H n,

8

.... ........

0

i ~ .....

HFll

1

.....

2

number density distributions were determined by various spectroscopic methods. In the axial region, electron number densities were determined from the width of the Hb line. For the plasma zone from 2.2 to 5.2 mm, a method based on the Saha jump was used [23]. For the calculation of the Saha jump, ion number densities in the ground state, as determined by absorption technique, were used [24]. Details of the measurement techniques used have been previously described [16,17]. In this work, measurements were extended to the fringe region. Beyond 5.2 mm distance, the electron number density was assessed from the total ion concentration, as determined by atomic absorption. For this reason, only analytes having ion lines accessible to conventional atomic absorption techniques were used. Since the ionization of the main gas components (Ar, O and H) is negligible in this region, the electron number densities were equated to the sum of the ion number densities of the corresponding analytes. The values obtained in this way represent the lower limit of the electron number densities. Results are shown in Fig. 8.

3

4

I .........

5

]~l .......

6

I .....

7

8

Radius, mm

Radius, mm Fig. 7. Radial distribution of the temperature. (x) from the Hb line width; (O) from the slope of the Boltzmann plot for emission; (Fq) from the slope of the Boltzmann plot for absorption; (*) from the intensity ratio of the Fe I line pair 373.487/373.713 nm;( + ) mean values from absorption of several Fe I line pairs.

ill,l~rlll,lll,~lFi

Fig. 8. Radial distribution of the electron number density. (O) from the Hb line width; (D) from the Saha jump; (*) estimated from the ion number densities of Ba and Fe.

4.4. Radial analyte distribution measurements To compare the computed distribution with the experimental data, experimental radial distribution of the analyte pressure is required. Radial distributions of analyte atoms and ions were derived from measured absorptions and integrated line radiances of the corresponding spectral lines for several analytes (Ba, Ca, Fe, Zn), and the pressures calculated according to Eq. (1). In selecting the analytes, the criteria of ionization energies spanning a wide interval as well as the availability of atom and ion lines in UV and VIS spectral regions, were emphasized. The pertinent data for these analyte lines are collected in Table 1. The ground or low-lying state populations (in m -3) were calculated from the measured quantities by using the relation:

A°°6~° np = 2.60.1014 0"939 fml~'Zq

(14)

and the excited level populations (m -3) by using the relation 3

Bx,cS)~il

nq = 9.489' 1029 )~Pqfqpl

lc'

(15)

89

M.S. PavloviO, M. MarinkoviO/Spectrochimica Acta Part B 53 (1998) 81-94 Table 1 Relevant data for used spectral lines Element

Ionization energy, eV

Wavelength, nm

Energy levels, eV

Ba

5.210

Ca

6.111

Fe

7.87

Zn

9.39

Ba I 553.584 Ba II 455.403 Ca 1 422.673 Ca II 393.367 Fe l 371.994 Fe II 259.940 Zn l 213.856 Zn 11 206.191

0-2.24 0 2.72 0 2.93 0 3.15 0-3.33 0-4.77 0-5.8 0-6.01

where A~o is the peak absorbance, 6XD (m) is the Doppler width of the absorption line, l(m) is the absorption path length,fpq andfqp are the absorption and emission oscillator strength, respectively, B~,c (in W m -3 sr -1) is spectral radiance of the standard source, ll and Ic are the measured line and standard source signal, respectively and 6X (m) is the spectral bandpass of the monochromator. The required number density of the species in all states, n T, is obtained with the help of the Boltzmann distribution formula:

nr = UnpeEp/kT, gp

(16)

where U is the partition function, Ep the energy of excited level, and k the Boltzmann constant. When the level is not the ground state, a factor containing the temperature in the exponent has to be used for transforming the data. This requires a very accurately determined temperature value. For this reason, we preferred absorption whenever the measurement was feasible. In some plasma zones, only one line from the required atom-ion line pair was spectroscopically measurable. For instance, at some positions close to the arc axis, only ion lines of Ba, Ca and Fe were measured, since the atom lines were either very weak or absent. For Zn, in the fringe, only atom lines could be detected. In these cases, the SahaBoltzmann equations were used to calculate the number density of the corresponding species. The analyte pressures obtained in this way are represented by circles in Figs. 9-12. In the fringe of the arc column, due to the low temperature, considerable fractions of some elements (particularly Ba and Ca) exist in molecular form, and therefore their total number densities in this region

gf 0.90 0.28 0.28 0.21 0.372 2.24 1.3 5.8

cannot be assessed only by atomic spectroscopic techniques. At the periphery of the arc column, r = 7.5 mm, a total number density fraction of analyte (neutral, ions and molecules containing analyte atoms) was estimated from the efficiency of the nebulizing system and the gas flow rate. A schematic diagram, showing the measured parameters and the various steps for the evaluation of the required quantities, is reported in Fig. 13. Measured quantities are given in the boxes on the left, arranged to match optimal temperatures in decreasing order, and two output quantities, the analyte pressure and the fraction of ionized analyte, are given in the boxes on the right.

4.5. Calculation of radial analyte distributions To exclude any doubt on the existence of LTE, values of X andpe determined by independent spectroscopic techniques were used to solve Eq. (10). The fraction of ionized analyte (X) was evaluated from the ratios of the analyte ion number density (n(~1)) determined spectroscopically to the total analyte number density, including the analyte-containing molecules (na =n(~°)+na(1)---('~)~-n~ )). For hot regions of the arc core, molecules were disregarded, while in the fringe region, n a was estimated from the nebulizer efficiency (that included analyte containing molecules). Eq. (10) is then solved by usingpa at 7.5 mm as the integration constant. It should be noted that the presence of the analyte molecules in the fringe of the plasma requires many additional parameters that are not included in the calculations. Therefore, the results for this region should be considered as only rough approximations. Since the experimental distributions in the arc core were considered more reliable, the calculated curves

M.S. Pavlovi6, M. Marinkovik/Spectrochimica Acta Part B 53 (1998) 81-94

90 -1.0

0.0

-2.0

-1.0

Z•-3.0

-3.0

-5.0

-4.0

........

0 Oo

Z -2.0

-4.0

-6.0 I

GO0

d .........

1

~ .........

2

b .........

3

f .........

4

i .........

i .....

5

6

"r'Er'''''"

7

-5.0

T , l , , , p

0

.....

1

it,,

I , .....

2

Radius, mm

r,,

I .........

3

I .........

4

I .........

5

I .........

6

~,

7

Radius, mm

Fig. 9. Radial distribution of the partial pressure of Ba. The solid curve is the calculated distribution; (O) experimental values.

Fig. 11. Radial distribution of the partial pressure of Fe. Solid curve is calculated distribution; (O) experimental values.

were shifted along the ordinate axis to fit the experimental points. Figs. 9-12 show the comparison of experimental values (circles) and calculated, monotonously rising, curves. The analyte depression at the arc axis, compared with the initial analyte pressure observed at the

periphery depends, as predicted by the model, on the ionization energy of the analyte atoms. For elements with low ionization energy, the analyte depression is very large, and on the arc axis amounts to about three-four orders of magnitude. However, the analyte pressure depressions calculated from experimental -1.0

-1.0

f

-2.0

-2.0

o

o

0

Z -3.0

E -3.0

Z

(A

M

-4.0

~ -4.0

-5.0

-5.0

-6.0

.........

i .........

1

~ .........

2

i .........

3

i .........

4

i .........

5

i .........

6

i .........

7

Radius, mm Fig. 10. Radial distribution of the partial pressure of Ca. Solid curve is calculated distribution; (O) experimental values.

-6.0

0

1

2

3

4

5

6

7

8

Radius, mm Fig. 12. Radial distribution of the partial pressure of Zn. Solid curve is calculated distribution; (O) experimental values.

91

M.S. Pavlovik. M. Marinkovi~'/Spectrochimica Acta Part B 53 (1998) 81-94

IJ~(T) , ~

P=nkT

!1

Boltz~nn eqn] I

r

:: n~ from H~

l i' Uc~°~(T) , ~ .

P=nkT

I Nebulizer efficiency Fig. 13. Schematicdiagramfor the evaluationof the requiredparameters,Xand P a, frommeasuredquantities--radiancesB and absorbancesA of relevant spectralatom and ion lines, and aerosoldensity--given in boxeson the left side. n, p and Xdenote particle numberdensity, partial pressure, and fraction of the ionized analyte, respectively.Subscriptsg and T refer to the ground state and to the sum of all states of relevant particle, and superscriptsdenote: 0, neutral; 1, singly ionized analyte; (*) excited state.

data, are smaller than those calculated by assuming LTE conditions. This could be ascribed to the fact that the experimentally determined electron number densities in the fringe of the arc column plasma are found to be higher by two orders of magnitude than those calculated from LTE for a similar arc plasma [22]. As a consequence, the corresponding gradients were lower, yielding less pronounced analyte depressions. For the elements with high ionization energy, the analyte depression becomes small or even insignific a n t - f o r Zn it is smaller than one order of magnitude (Fig. 12). This means that elements with high ionization energy can more easily reach the hot axis zone. Experimental values agree well with calculated curves for plasma zones close to the axis. However, in the fringe zone, an experimentally determined decrease in pressure was observed for some analytes. Such a drop could be related to the formation of stable molecules, due to the low plasma temperature in this region. The largest drop, observed for barium, could be due to the formation of stable barium hydroxide molecules. In the fringe region, the same mechanism could be a possible reason for the absence of the steps that are apparent in the calculated partial pressure curves in Fig. 4 and Fig. 5.

5. Discussion 5.1. Restricting assumptions

In developing the mathematical model (Eq. (9) and Eq. (10)), many restricting assumptions have been adopted (1-10 in Section 2). Many of them are obvious but three of them will be discussed here: 1. The assumption that the temperature is not appreciably influenced by the presence of the analyte is satisfied only when the analyte fraction is small. In spectrochemical plasma sources, samples are introduced as aerosols, usually generated with pneumatic nebilizers. The total amount of dissolved solid, due to restrictions imposed by nebulizers, is usually held below 0.25% by weight. For analytes of relative atomic mass of 50 and nebulizer efficiency and gas flow rates as in Ref. [ 17], the analyte number density fraction in plasma amounts to 3.10 -5. This small analyte fraction, even if completely ionized, cannot appreciably increase the electron number density in the arc core, where the bulk of current flows, and therefore, the bulk of the heat is generated. Heating of the fringe region by conduction and therefore its

92

M.X Pavlovik. M. MarinkoviO/SpectrochimicaActa Part B 53 (1998) 81-94

temperature, where the demixing mainly occurs, remains unchanged. 2. A hypothetical fill gas, with an ionization energy of 14 eV, in the temperature range defined by Eq. (1 l), provides a degree of ionization that is close to the resultant (effective) ionization of the gas mixture of argon, hydrogen and oxygen (from water vapor that originates from nebulized solutions) found in the real d.c. plasma [17]. 3. The partition function ratios (Eq. (2)) are usually either larger or smaller than unity, but for some elements with complex spectra they are close to unity. The ratio equal to unity, as used for modeling (Section 2 and Section 3), makes clear the dependence of analyte pressure on the analyte ionization energy, since this dependence was not distorded by various partition function values. For experimental testing (Section 4). the exact partition functions for corresponding temperatures are used. However, in spite of the imposed restrictions and simplifications, the modeled plasma conditions are not far from those of the real d.c. plasma sources used in emission spectrochemical analysis.

of both absorption and emission measurements, made the requested measurements possible. 5.3. Static atmosphere approximation

How much the arc plasma used in this work conforms to the static atmosphere assumption could be estimated by comparison of the gas velocity flow with the diffusion velocity of the analyte. From the gas flow and the cross-section of the channel (Fig. 1, in Ref. [17]), the axial gas velocity of 7.5 cm s -l (at room temperature) or 70 cm s -l (at a mean gas temperature of 3000 K) were obtained. The gas, introduced tangentially into the cavity of the central segment, retains non-turbulent motion along the channel. It is believed that the gas velocity perpendicular to the arc axis is much smaller than in the axial direction. Due to the high temperature gradient in the arc plasma, a considerable contribution by diffusion is expected. The velocity of diffusion in the radial direction is given by: V = - D d(ln n) dr

5.2. Spatial distribution measurement

A specific feature of the plasma source used in this work is that it allows to perform, in addition to the commonly used side-on observation followed by an Abel inversion procedure, direct measurement of the radial distributions by end-on observation. The Abel inversion procedure suffers from two majority drawbacks: (a) it requires good axial symmetry; and (b) an inherent error propagation deteriorates the reliability of the calculated values in the axial region. The above disadvantages are particularly severe when the variable to be calculated has off-axis peak values. In that case, the accumulated measurement error and the error due to a deviation from the perfect symmetry condition may become large enough to mask the measured quantity. The Abel inversion procedure can be hardly applied for spectroscopic plasma sources if depression at the axis attains 90% of the off-axis peak. In the present work, we had to deal with depressions at a plasma axis as large as four orders of magnitude. The effectively eliminated plasma plumes at the end of the observation path and the combination

'

(17)

where D is the diffusion coefficient, and n the analyte number density. The magnitude of the diffusion coefficient has been estimated in many papers [25,26]. As a rough estimation, a mean value of 15 cm 2 s -L was taken for elements in the middle of periodic system. The magnitude of the analyte concentration gradients was estimated from the experimentally determined radial analyte distributions. It may attain a value up to one order of magnitude per milimetre. If this value is entered into Eq. (17), a diffusion velocity of 3.5 m s -L is obtained. From the above considerations, it follows that the gas flow velocity in the radial direction is much smaller than the diffusion velocity. The assumption of a static atmosphere is then reasonably fulfilled. 5.4. Reliability o f measured and calculated data

The slope of the electron number density is strongly correlated with analyte spatial distribution (Eq. (10)). The greatest change of the number density fraction for analytes of low and medium ionization energies

M.S. Pavlovi?, M. Marinkovi?/Spectrochimica Acta Part B 53 (1998) 81-94

occurs in relatively cool plasma regions beyond 2.5 m m distance from the axis. Unfortunately, there are no reliable methods for electron density determination in the density range involved. The most accurate method (Hb line width) was not applicable in this region because o f low intensity and insignificant Stark broadening. Since the ionization o f the main gas components at low temperatures is negligible, the electron number densities beyond 5.2 m m were assessed from the analyte ion number density, experimentally determined by atomic absorption. Equalizing the electron to ion number densities is possible since the existence o f negative ions is not likely in the plasma investigated. In spite o f many simplifications in the model and the uncertainty o f some measurement data, the predictions o f the model, particularly about the drop o f the analyte pressure at the arc axis, are found to be in good agreement with the experimental findings. Indeed, the effect is so large that it can be considered as unambiguously verified. The drop in analyte pressure or analyte number density in the arc core zone, following earlier papers [13-15], was qualitatively interpreted as a consequence o f the repulsion o f the aerosol particles by the plasma due to the high temperature gradient. However, since with our model the effect is quantitatively explained by the radial electric field, it is proposed here that, instead o f the widely accepted term "thermal barrier", the term "potential barrier" would be more appropriate and should, therefore, be adopted.

6. Conclusions A one-dimensional model was proposed to predict an aerosol sample introduction efficiency by the interaction o f the radial electric field with the partially ionized a n a l y t e - - a n element with low ionization energy and present at low concentrations. The calculations for the close-to-real arc plasma showed the strong dependence on the ionization energy and concentration o f the analyte. The depression in the analyte pressure at the arc axis may exceed four orders o f magnitude. Experimental tests, carried out with Ba, Ca, Fe and Zn as analytes, showed a good agreement between the experimental and the modeled distributions.

93

In the past, the effect has been described by the term " t h e r m a l barrier"; however, from the results o f this work, the term "potential barrier" seems more appropriate.

Acknowledgements The authors wish to express their thanks to Prof. B. S. Mili6 for his valuable advice during the preparation o f the manuscript.

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