Anal. Chem. 1987, 59, 2861-2872
2861
Theoretical Examination of Solute Particle Vaporization in Analytical Atomic Spectrometry Gary M. Hieftje,* Richard M. Miller,l Yongnam Pak, and Erland P. WittigZ Department of Chemistry, Indiana University, Bloomington, Indiana 47405
Two atternatlve models are developed and compared for the vaporlratlon of slngle alkall chloride particles In analytical flames or plasmas. Mathematlcal expresslons are derived from both models whlch enable -the calculatlon of partlcle vaporlration rates or vaporlratlon tlmes from obtalnable physical parameters. From these rates and times, It Is concluded that the volatlllzatlon of large (>lo pm) alkall chloride partlcles should be controlled by the rate at which heat can reach the partlcle surface. I n contrast, the evaporatlon of small solute spheres Is rate-llmlted by molecular release at the particle surface, a sltuatlon slmllar to that encountered durlng vacuum evaporatlon. Nelther model, however, can successfully predict the entlre hlstory of a large solute mass whose vaporization traverses both reglmes. Instead, theory and experbnenl both reveal a change In volatlllratlon behavlor as the partlcle reaches submlcrometer dlmenslons. Exlstlng theories cannot successfully predict just where this break In behavior wlll occur. Fortunately, most solute particles encountered In practlcal analytlcal atomic spectrometry wlll fall In the small-partlcle slre range, so their vaporization wlll proceed vla a mechanism In whlch the partlcle radlus decreases llnearly wtth tlme.
Despite the acknowledged and widespread utility of flame and plasma atomic spectrometry, surprisingly little is known about the processes whereby an aerosol droplet is converted to free atoms or ions used in the determination. Intuitively, one recognizes that such a droplet must undergo desolvation to produce a solute particle (or particle cluster) which might subsequently melt, fragment, decompose, or sublime. From these events, vapor-phase species will be formed which will include atoms, ions, molecules, and molecular fragments, all apparently in rapid equilibrium. Of these various steps, the conversion from solid solute to free atoms or ions is the most important, for it is during this step when most interelement “interferences” arise. In flame spectrometry, such interferences result in the well-known depression of certain analyte signals (e.g. Ca) by specific concomitants (e.g. phosphate or aluminum). In some forms of plasma spectrometry, also, vaporization interferences are not uncommon; in other plasma methods the production of ions, especially from an easily ionized element (EIE), is of importance and dominates interelement effects. An understanding of the processes and kinetics of particle vaporization in flames and plasmas would allow calculation of the fraction of a sample that is released into the atom cloud under particular experimentalconditions. From the calculated residence time in the flame or plasma, and the characteristics of the sample, it would then be possible to compute the maximum particle size which could be completely vaporized by the time it reached the observation zone. Such information could be useful in optimizing torch and nebulizer design to improve the performance of practical atomic spectrometers. Permanent address: Unilever Research, Bebington, Wirral, Merseyside L63 3JW, U.K. Present address: Chevron Research Co., Richmond, CA 94802. 0003-2700/87/0359-2861$01.50/0
Past studies of vaporization and ionization in atomic spectrometryhave been largely empirical and have been aimed at eliminating interference effects of the kind just noted. However, in other studies, attempts were made to characterize these events on a more fundamental level. Unfortunately,such studies usually relied upon conventional sample-introduction methods and were therefore limited to observations on a complex aerosol in which droplets of various sizes interacted and traveled in an uncontrolled way. To overcome these limitations, recent studies have focused on the behavior of individual aerosol droplets and particles injected into analytical flames. In such studies, the vaporization of individual particles could be examined (1)and the liberation of atoms from them monitored (2-4). By use of a simplified model of such vaporization, ionization could be studied as well (5). In the present study, an overview of theoretical models of the vaporization process in analytical flames and plasmas will be presented. Alternative models will be compared with experimental data, and limitations and difficulties in applying current theories will be evaluated. For simplicity, modeling will be restricted to the alkali halides. These systems are easiest to study because the necessary physical data required for modeling is often available and because vaporization anomalies are less frequent (I). In addition, no allowance will be made for the effects of ionization. Although the actual ionization process is easy to model, ion-electronrecombination complicates the treatment substantially. The effects of temporal and spatial variations in an expanding atomic-vapor cloud produce similar changes in the kinetics of the two-body ion-electron recombination event and cause a situation where ionization effects cannot be reliably included in the model. Conveniently, vaporization-rate data have recently been obtained for alkali halides in flames in which ionization has been suppressed by the addition of easily ionized cesium atoms (4). Consequently, direct comparisons can be made between the various models discussed in the present paper and the analytical data obtained earlier. All of the analysis and modeling presented here refers to the vaporization of indiuidual particles. Extension to clouds of particles and complete aerosols is possible by using the approach developed in Li (6). VAPORIZATION MECHANISMS As shown by Bastiaans (1)and Skogerboe (3,particle vaporization in analytical flames and plasmas can be extremely complex, especially when mixed or reactive solutes are examined. In some cases, mixed solutes can vaporize sequentially, with the earlier-volatilizing component serving as a carrier of the more refractory one. In addition, vaporization of large solute masses can occur in part by fragmentation, perhaps induced by entrapped traces of solvent (1). Presumably, reactive solutes can produce a similar phenomenon. Even for relatively simple materials, vaporization can be preceded by the formation of a rather plastic substance, capable of being blown into “microballoons” (7). Conveniently, some solutes, such as simple oxides and alkali halides of interest to the present study, vaporize in a relatively straightforward manner. For such materials, the volatilization process can be kinetically controlled by either of two events, 0 1987 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 59, NO. 24, DECEMBER 15, 1987
for those such as the alkali chlorides which melt well below common flame or plasma temperatures. In some situations, it is convenient to express eq 1in terms of the rate at which solute mass is lost by a vaporizing particle. This alternative expression can be obtained by differentiating eq 1 and equating solute radius and mass ( m )by use of the particle density ( p ) . Equation 1 gives 2r dr = -kvl d t (2) and the change in mass of the vaporizing particle (dm) is d m = 4irpr2 dr
(3)
The desired expression is therefore -dm/dt = 2arpkVl F@m 1. Schematic representation of a vaporlzlng particle. Important fundamental constants and temperatures Include the following: A,, thermal conductivity of flame gases; A,, thermal conductivity of diffusion mixture; A,, thermal conductivity of solute vapor; D , diffusion coefficient of vaporlzlng species; T,, surface temperature of particle: T,, flame gas temperature.
both of which can be understood with the aid of Figure 1. In Figure 1,a solute mass or its molten counterpart is surrounded by a thin film of vaporization products which diffuse away from it. In this model, the solute material has achieved the velocity of the flame or plasma gases (8), so that it can be viewed as residing in a quiescent environment. In this situation, the rate of evaporation of the solute might be governed either by the rate at which energy (heat) can be transported to the solute surface or, alternatively, by the rate at which solute material can leave that surface. These cases are termed heat-transfer or mass-transfer limiting, respectively. The heat-transfer case would be controlled by physical parameters such as the heat capacity of the volatilized solute material and by the thermal conductivity of the medium surrounding the particle. The limiting thermal conductivity might be either that of the thin vapor film or of the flame gases, whichever is less (9). In contrast, mass-transfer-controlled vaporization would be limited by such parameters as the diffusion coefficient of the vapor leaving the particle surface and the vapor pressure of the volatilization products at the particle-surface temperature. Intuitively, one would expect the particle volatilization rate to be limited by heat transfer for low-boiling solutes. For such materials, the surface temperature would of necessity lie far below that of the flame or plasma gases. The resulting steep temperature gradient a t the particle surface would strongly suggest a heat-transfer limit. The same process reportedly governs the desolvation of droplets in high-temperature flames (9).
Mass-transfer control, on the other hand, would be expected for high-boiling (refractory) solutes or those which sublime. This behavior would be similar to that observed in the classical studies of Langmuir in his investigations into the evaporation of iodine spheres (10).The situation is similar also to that experienced in the evaporation of aqueous aerosols in spray chambers employed in analytical flame and plasma spectrometry ( 2 1 ) . Conveniently, both heat-transfer-controlled and masstransfer-controlled vaporization follow the same rate expression for the evaporation of large particles: r2 = ro2- kvlt (1) In eq 1,the particle is viewed as being spherical of radius r and initial radius r,,. The vaporization rate is kvl and the expression shows that the particle surface decreases linearly with time ( t ) . Although the assumption of a sphere might not be valid for the vaporization of some solutes, it should hold
(4)
An interesting conclusion results when eq 4 is recast in terms of the rate of mass loss per particle area ( A ) . This relationship, shown in eq 5, reveals that rate of mass lost per 2r unit area is inversely proportional to particle radius. As a consequence, the mass flux per unit area becomes unrealistically high as the particle becomes extremely small, suggesting that another process must become limiting. In fact, this alternative process is the rate a t which vaporizing species are able to leave the particle surface and is similar in behavior to vaporization under vacuum conditions (22). This behavior is governed by an alternative rate expression r = ro - kV2t (6)
A change in particle volatilization behavior from heattransfer control to mass-transfer control can be understood qualitatively in terms of the relationship between particle size and the mean free path of molecules leaving the surface. The key to understanding this change arises from consideration of the thermal gradients surrounding the particle. During heat-transfer-controlled vaporization, a relatively large thermal gradient must exist around the particle; without such a gradient, there would be no drive for a heat-transfer process. Conversely, under mass-transfer-controlled conditions, no such thermal gradient can exist, since by definition the rate of mass loss by the particle, and not the heat gain, becomes controlling. Because heat can then reach the particle a t a more than sufficient rate, there is no temperature gradient until one reaches within a distance from the surface approximating the mean free path. This argument was proposed earlier by Clampitt and Hieftje (9) and more recently by Chen and Pfender (23-16). These latter authors defiie this region where the gradient exists as the “temperature jump distance”. However, the two views are conceptually the same. Consider the two limiting situations. For a large particle, mass lost by the particle per unit solid angle consumes a great deal of heat. Therefore, during its vaporization, the particle depletes the heat content from its surroundings and thereby establishes a temperature gradient. Heat transfer through that gradient region then limits the particle volatilization rate. In contrast, consider an extremely small particle. Per unit solid angle, that particle requires very little heat for vaporization. As a result, the particle does not act as a thermal sink and the vapor surrounding the particle reaches the gas (flame or plasma) temperature. A temperature gradient can then exist only within a few mean free paths of the surface. Under these conditions, particle vaporization can be viewed as being limited by either of two factors: the rate at which the surrounding gas molecules transfer energy to the particle through that extremely thin region, or the rate at which molecules can be released from the particle surface. In the model of Chen and Pfender (13-16), who choose the former of these concepts,
ANALYTICAL CHEMISTRY, VOL. 59, NO. 24, DECEMBER 15, 1987
it is the "Knudsen effect" which limits the rate of heat transfer to the particle surface. The Knudsen effect describes the collision rate and effective energy transfer efficiency between external molecules and the particle. In the molecular-release model, which is actually no different, vaporization is dictated by the surface temperature of the particle and the efficiency with which molecules can leave the particle surface. In fact, both models employ the same parameters and incorporate in them a semiempirical factor called the "thermal accommodation coefficient" in the heat-transfer model or the "evaporation coefficient" in the molecular-release model. The principal distinction between the two models appears to be whether one considers continuously heat transfer into the particle or conceptually alters the view from heat-transfer to mass-transfer control. Although the former view might seem to be more consistent, the latter is easier to understand, especially if one a!ters the model so that the thermal gradient around the particle is destroyed entirely. That is, the particle surface itself warms up to the temperature of the surrounding gases. Such a situation is possible, even if the particle's boiling point lies well below the temperature of the gases, if molecular release at the particle surface limits the rate at which the particle can evaporate. Under these conditions, sufficient thermal energy is available for the particle to vaporize, but the molecules cannot be released from the surface at a high enough rate. The particle can therefore be considered to be in a "superheated" state where the molecules composing the particle experience an overpressure relative to the surroundings due to their greater kinetic energy. In the following sections more quantitative expressions for the concepts introduced above will be described and evaluated. Heat-Transfer-Controlled Vaporization. As shown in Figure 1, the transfer of heat to a particle might limit its rate of vaporization. Equation 7 represents such an energy-transfer process (17). The left-hand term represents the rate of
4ar2h,(Tg- T,) =
$( $)
+ 4ar2t,aT,4
(7)
conductive heat gain by the particle from the flame or plasma, whereas the right-hand side indicates the distribution of the gained energy. The first term on the right side denotes the energy consumed by species during vaporization from the particle surface and the second term defines the amount of energy that is lost by radiation from the hot particle. In eq 7, h, is the heat-transfer coefficient (described in detail below), Tgis the temperature of the flame gases, T,is the temperature of the particle surface, AHv is the heat of vaporization of the analyte salt, M is the molecular weight of the vaporized species, dmldt is the change in the particle mass with time, t, is the emissivity of the molten salt, and a is the StefanBoltzmann constant. (A complete glossary is included at the end.) Other workers (18)include in this heat-transfer equation terms for radiative heat gain by the particle. However, being in radiative disequilibrium, analytical flames and plasmas contribute little to the process of radiative heat transfer to a particle; the spectral regions of strong flame background emission do not coincide with regions where the particle absorbs significantly. Moreover, radiation emitted by the molten salt particle should not influence other particles because the particles are well separated in space. The heat-transfer coefficient in eq 7 is defined (18) as
where X is the lesser thermal conductivity-either that of the
2863
flame or of the analyte vapor, AHov is the overall heat of conversion necessary to take the vaporization products from the particle surface temperature (TJto the flame temperature (Tg),and A (the bracketed term) is the mass counterflow coefficient. Nu is the dimensionless Nusselt number which enables a correction to be made for convective heat addition to the particle. The Nusselt number can be expressed in terms of the Prandtl (Pr) and Reynolds (Re) numbers
+
Nu = 2 0.6Pr0.33Re0.5 (9) Because the particle velocity is very nearly the same as that of the flame or plasma gases (8), the Nusselt number is 2.0 here (18). Substituting eq 3 and 8 into eq 7 , one obtains -A",4.rrpr2 dr - 4ar2tE,aT,4 (10) 4arXA(Tg- T,) =
M
dt
+
which can be rearranged and integrated
M
J r
r dr (11) XA(Tg- T,) - rt8aT,4
where at t = 0, r = ro (initial particle radius). Upon integration, eq 11 becomes
t=
-[
M V P
r-ro+
Me,aT: XA(T, - T,) -rt,aT,4 XA(Tg- T,) - rotsaT,4 If the term involving radiative heat loss is small, eq 12 becomes simpler to handle. Conveniently,radiative heat loss can be neglected for alkali halide particles (and probably for those of most other common solutes). Substitution into eq 12 of the values pertinent to alkali halide vaporization (cf. Table I) shows that radiative heat loss does not significantly contribute to the time it takes a particle to vaporize. The time required for the complete vaporization of a sodium chloride particle of 0.58-pm diameter can be calculated with eq 12 and the data in Table I to be 5.13 X s. If heat loss is neglected, eq 14 (discussed below) holds and yields a value of 5.06 X s. The small difference ( 1%)suggests that radiative heat loss is indeed a minor factor. Deleting the radiative heat loss term in eq 10 yields N
4arXA(Tg- T,) =
-AHv4apr2 dr M dt
(13)
Upon integration, eq 13 becomes
t = A",p(ro2
r2) 2MAX(Tg-Ts) -
which can be recast in the form of eq 1
2MAX(Tg- T,) (15) In eq 15, the bracketed term can be viewed as the vaporization rate constant kVl. The Knudsen effect predicts that as the particle size decreases, the rate of heat transfer to the particle surface will be reduced more rapidly than would be predicted by the decrease in surface area alone. This effect arises because the probability of successful energy transfer from the flame or plasma species to this particle is diminished as the particle dimensions become small with respect to the mean free path of the gas species (14). The importance of the Knudsen effect can be assessed in terms of the Knudsen number K ,
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ANALYTICAL CHEMISTRY, VOL. 59, NO. 24, DECEMBER 15, 1987
K, = L/2r
(16)
where L is the mean free path of the gas species and r is the particle radius. For K, < lo4, the Knudsen effect is negligible and simple continuum heat-conduction calculations can be used to determine the heat flux to the particle. For K , > 10, continuum concepts become inapplicable, and mass-transfer considerations based on the kinetic theory of gases apply. At intermediate Knudsen numbers, heat-transfer calculations need to take account of the Knudsen effect. Chen and Pfender (16)have shown how heat-transfer calculations can be modified to include the Knudsen effect. They calculate the true rate of heat transfer Q to the particle from the calculated continuum rate Q, with the equation
where r is the particle radius and Z* is the effective temperature-jump distance. Q, corresponds to the left-hand side of eq 7. Chen and Pfender (16)found that eq 17 is valid for < K , < lo-'. For lo-' < K , < 10 there is a transition region between the temperature-jump regime and the masstransfer regime which is not easily treated theoretically. However, they reported experimental evidence that eq 17 could be applied to situations in which the Knudsen number reaches at least 0.8; the applicable range of eq 17 to larger K , might be valid but cannot be fundamentally justified at present. As will be shown later, the vaporization of typical solute particles in a flame or plasma pertains to K , > 1. However, because no alternative theoretical treatment presents itself, we have assumed that eq 17 can be applied to the entire history of the particles under consideration here. The temperature-jump distance Z* is the range over which the temperature of the system changes from that of the particle surface to that of the ambient gas and defines the slope of the temperature gradient in the vicinity of the particle. The value of Z* depends on a number of parameters including the Knudsen number. The effective temperature-jump distance Z* is given by
Z* = or
r =
i)( (?)( i)( (?)(
l + Y
&)L*
l + Y
k)K,*
(18)
(19)
where a is the thermal accommodation coefficient, y is the specific heat ratio, Pr is the Prandtl number, L* is the effective mean free path, and K,* is the effective Knudsen number under plasma conditions. The thermal accommodation coefficient (a) defines the effectiveness with which kinetic energy from gas species impinging on the particle's surface is transferred to the particle and is given by a=-
Ei - E, Ej
- E,
where Ei is the energy carried to the particle surface per unit area and unit time by the incident gas species at the gas temperature, E, is the energy retained by the gas species after impact, and E, is the energy that they would be expected to retain if they reached equilibrium with the particle surface temperature. The final parameter which must be calculated in order to assess the importance of the Knudsen effect is the effective mean free path L* of species leaving the particle. The conventional concept of mean free path does not apply to situations involving distances less than 1 ym (19). In the theory
of transport to or from particulates, the mean free path should be treated not as the mean distance traveled by a molecule between consecutive collisions but as the mean effective free path L* which is, for a mass-transport process, the mean distance that a molecule has moved from its initial position by the time its velocity vector becomes independent of the initial velocity vector. This concept becomes more important when the mass ratio of vaporizing gas molecules (m,) to the surrounding gas molecules (mg),m,/m,, is large. When the scattering of vapor molecules is anisotropic and their velocities show a persistence after collisions, the probability of their returning after the first collision evidently decreases, and this decrease is more pronounced for stronger persistence (i.e., for higher values of mv/mg).Here, mv/mgvaries from 1.5 for LiCl and 2.1 for NaCl to 2.6 for KC1, assuming the surrounding gas to be comprised principally of N2. Jeans (20) derived the relation for L*
L* = L(m,/mg + 1)/(1
+ 6)
(21)
where L is the mean distance traveled by the vapor molecules between collisions (the conventional mean free path) and 6 is a diameter-related parameter that is zero for m,/m, = 0, slowly increases with increasing mv/mg,and reaches 113 for mv/mg= a. Here, 6 is chosen as 0 because it increases slowly with the mass ratio m,/m, which is about 2 for NaC1. The final expression for L* is (1
L* =
+ m,/mg)1/2
(1 + 6)angdP2
(22)
where the subscripts v and g refer to the vaporizing molecule and the surrounding gas, respectively. In eq 22, ng is the number density of the surrounding gas obtained from the ideal gas law equation, d is the molecular diameter, and d, = (1/2)(dv dg). The temperature of the vapor surrounding a particle is assumed as being that of the particle's surface. An equation similar to (22) is used by Chen and Pfender (14) to derive the effective mean free path. From eq 17, whenever the ratio Z*/r is small, the heat flux to the particle approaches the continuum limit. This situation will occur for small Knudsen numbers, that is, whenever the particle radius is large or the effective mean free path is small. For small particles, the Knudsen effect becomes progressively more important and reduces the rate of heat transfer as Z*/r becomes larger. In the limit, the effective rate of heat transfer becomes zero and vaporization is solely controlled by masstransport considerations. Mass-Transfer-Controlled Vaporization. Evaporation of liquids under mass-transfer control is the basis for the postulate that vaporization of analyte particles in a flame might occur by the same process. Langmuir (10) modeled the evaporation of iodine spheres as a mass-transfer-controlled event. However, the model's prediction deviates from experimental observations as the evaporating sphere becomes small compared to the mean free path of the liberated molecules. This fact, suggested above, led Bradley et al. (21) to modify the mass-transfer-controlledvaporization expression. More recently, Monchick and Reiss (12) combined these two equations into the following:
+
where v is the velocity of particles striking (or leaving) the particle surface v = (RTg/27rM)'1'
(24)
In eq 23 and 24, a is the "evaporation coefficient" (211,P, is
ANALYTICAL CHEMISTRY, VOL. 59, NO. 24, DECEMBER 15, 1987
the vapor pressure of the volatilized solute at the particle surface, R is the gas constant, and D is the diffusion coefficient of the vaporized material. The last term in the denominator of eq 23 governs whether “small particle” or “large particle” vaporization behavior ( I , 5 ) will exist. If the particle is small, the product avr > (1 - a / 2 ) D ,their ratio is much greater than unity, and eq 23 becomes
which governs large-particle behavior. Not surprisingly, eq 25 and 26 are identical with those derived by other workers to describe explicitly small particle (21,22)and large particle (IO) vaporization behavior. Equation 26 can be integrated over the same limits as eq 11 to give eq 27. Again, eq 27 has the same form as eq 1, with the bracketed term being the “large particle” vaporizationrate constant kVl.
r2=r:-
[
2MP,D (27)
Similarly, eq 25 can be integrated to produce eq 28.
However, eq 28 is not of the same form as eq 1, but instead suggests, for small-particle vaporization, a linear decrease in particle radius with time (cf. eq 6). Because of this difference, the bracketed term in eq 28 will be designated the smallparticle vaporization rate constant kv2. Equation 28 can be expected to be important in the vaporization of small particles, even if the Knudsen effect is included in the heat-transfer model. The reason is that in the “molecular free-flow” regime (K,, > l o ) , heat transfer is not limiting (14).Unfortunately, in the heat-transfer model the entire vaporization history of a particle cannot be treated by one equation such as (23);instead, some combined equations involving a crossover of limiting mechanisms would be required. In contrast, the vaporization of large particles by masstransfer control can be treated throughout the particle history by eq 23. Integrating eq 23 provides the general relationship for mass-transfer-controlled volatilization t =
L
Equation 29 cannot be cast in a form similar to eq 1. Not surprisingly, eq 29 reduces to the small-particle (eq 28) or large-particle (eq 27) relation, respectively, if its f i t or second term is used. Modeling of Vaporization Processes. In order to predict the behavior of a vaporizing particle during its lifetime and to compare the various models and the effects of different solute, flame, and plasma parameters, a computer simulation of the vaporization process has been carried out. In this simulation, the entire history of a vaporizing particle can be examined, whether it follows heat-transfer control governed
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by eq 7 , when that model is modified by eq 17 to include the Knudsen effect, or where mass transfer governed by eq 23 prevails. The use of a computer simulation is possible even when analytical solution of the pertinent relation is impossible; in addition, the approach allows convenient graphical representation of the predicted vaporization behavior and easy manipulation of the various parameters in the equations. Each model was programmed in FORTRAN-77 on a Control Data Corp. CYBER 170-730 computer using the conventional finite difference formulation of the heat-transfer problem (23). Simulated time is divided intosmall increments At. For each time increment, the change in particle radius or the change in particle mass is calculated from the appropriate governing equation. The new particle dimensions are then computed and used for the next time step. Providing the time steps are sufficiently small, the entire history of the particle can be determined to a good approximation by iteration. Double-precision arithmetic was used throughout to avoid the accumulation of truncation or rounding errors. For convenience in analysis, the model can display the history in terms of radius or mass changes. Application of Vaporization Rate Expressions. To test the models represented by eq 15 and 17 and 27-29 requires a means of measuring the vaporization kinetics of a single particle, the availablility of values for the physical parameters present in each expression,and a knowledge of the conditions under which each expression is valid. The measurement of particle vaporization rates is possible by using techniques which have been previously described (1-5,8, 9). In this section, values for the physical variables will be presented and the range of application of each expression will be explored. Physical constants required to apply eq 15, 17, 27,28, and 29 to the vaporization of several alkali-metal chloride particles are compiled in Table I. It will be noted that not only the melting point but also the boiling point of each of these salts lies well below the temperature of common analytical flames and plasmas. Consequently, it can be safely assumed that each vaporizing solute mass will be molten and therefore spherical. In addition, because vaporization proceeds at atmospheric pressure, the surface temperature of the molten sphere (T,) will be at the boiling point of the respective solute and the saturation vapor pressure of the solute at the surface (P,) will equal 1 atm (except for KCl, which sublimes). Moreover, the disparity between the flame temperature (T,) and the molten droplet surface suggests a strong temperature gradient near the solute surface, a situation that argues for heat-transfer control (9). The temperatures at which the constants in Table I have been evaluated in some cases deserve further comment. The heat capacity of solute vapor (c,) has been calculated at the flame temperature (T,) since the solute vapor will eventually be heated to that level and because heat capacity increases with temperature. Therefore, the quantity of heat ultimately consumed in warming the vapor will be represented by the value of cp at the vapor’s highest temperature. The thermal conductivity of the vapor (A) has been evaluated at T,. One could argue also that X should be taken at the gas temperature Tgfor reasons that were cited earlier (35). However, the overall rate of heat transfer is likely to be determined by the conductivity at the particle surface, because it is slower and therefore rate-limiting. Because the density ( p ) used in the earlier expressions is employed to convert particle mass into its radius equivalent, it has been cited at the particle boiling point (TB). For parameters such as thermal conductivity (A) and heat capacity (c,) to be useful in this treatment, it is clearly important to determine the species to which these parameters should pertain. Here, the species are those which immediately
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ANALYTICAL CHEMISTRY, VOL. 59, NO. 24, DECEMBER 15, 1987
Table I. Physical Parameters Required for Calculation of Vaporization Rate Constants for the Alkali Chlorides parameter AH" CP
x M TB
T8
units cal/mol cal/(K mol) cal/(cm s K) g/mole K K
P
g/cm3
D
cm2/s
P 8
R ea U
dyn/cm2 (dyn cm)/(K mol) cal/(cm2s K4)
P E Mdo MI0
Mo; Mo"g
kcal/mol kcal/mol cal/mol kcal/mol
h 118
L'
cm
for NaCl
40 808 9.370 9.08 x 58.44 2440 1738 1.244 9.9 1.013 X lo6 8.16 X lo7 0.9 1.30 X lo-'* 1 0.2 98 118.5 1.28 x 105 1.04 x 105 0.453 0.497 3.07 x
for LiCl 35 960 9.25" 1.37 X 42.39 2440 1655 1.159 8.0
same same 0.9
same 1 0.1 113 124.3 1.33 x 105 1.20 x 105 0.418 0.440 3.14 X
surround a vaporizing particle and have been assumed to be the respective alkali chloride molecules. Fotiev and Slobodin (36)have found that the intermolecular bond breaks during the sublimation of NaCl. Moreover, Kvande (37) determined from the molar mass of the vapor above molten NaCl (1293 K) that approximately 38% of the vapor consists of dimers, the balance being NaCl molecules. At the higher temperature of vaporization and in the region around a vaporizing particle, it is likely that most of the species will exist as monomers. Importantly, for monomers of alkali-metal chlorides, the thermal conductivity lies below that of common flame or plasma gas mixtures (9),making it the limiting parameter for heat-transfer-controlled vaporization. Accordingly, the thermal conductivity values in later calculations were those of the respective alkali chloride molecule. The parameter AHovrepresents the overall enthalpy required to take a vaporized alkali chloride molecule at the particle surface temperature to vapor-phase species at the flame temperature. Consequently, AHovcan be taken as the sum of the energy required to raise the respective molecule to the flame temperature plus the energies consumed by partial dissociation of the molecule and by partial ionization of the resulting atoms. In turn, the heat necessary to raise the vapor to the flame temperature is just the product of the vapor heat capacity and the temperature differential between the particle surface and flame. The heat involved in dissociation has been taken as the product of the molar heat of dissociation of each molecule ( A H d " ) and the fraction dissociated (8). Similarly, the energy consumed by ionization was calculated as the ionization enthalpy (AHi")times the fractional ionization (6). However, when ionization is totally suppressed, AHi" is not included in calculation of Hov. Perhaps the greatest uncertainty in this theoretical treatment lies in assigning a value to a , the "vaporization coefficient" required in eq 28 and 29. For vaporization of NaCl from large surfaces just above the melting point, Ewing and Stern (38)determined a empirically to be 0.23. However, for evaporation near the boiling point, complex molecules (such as ethanol and water) exhibit vaporization coefficients as low as 0.02, according to Hirth (39). The importance of these extremes in a will be investigated in a later section. For calculations involving the Knudsen effect, values for the main parameters were drawn from Chen and Pfender (14). The specific heat ratio y was taken as 1.40 and the Prandtl number Pr was assumed to be 2.28, corresponding to a flame whose principal constituent is nitrogen. The thermal accom-
ref and notes
for KC1 38 840 9.43b 5.28 X 74.56 2440 1680 1.157 5.4 9.53 x 105e
24 "25 b26 c25 dcalcd 27 24 28, 29 30
e31
same 0.9 same 1 0.6 100.5 100.1 1.67 x 105 1.07 x 105 0.388 0.480 3.11 X lo4
assumed 32 32 33 34
calcd fionizationincluded in calculatin gionization not included calcd gionization not included calcd
modation coefficient (a) was assumed to be 0.8. There is little experimental justification for assuming a value of 0.8 for a, except that it has produced good fits between calculated and experimental data (14). There is some evidence that for evaporating particles with clean surfaces, it might be somewhat lower, perhaps as low as 0.1 (14, 40). In such a case, the Knudsen effect would be stronger, although the form of the equations would be the same. The thermal accommodation coefficient must therefore be viewed with the same suspicion as the evaporation Coefficient CY until further data are available. The effective mean free path L* was calculated by use of eq 22. The flame molecular diameter d, used was that of N2 which is 1.098 A and d, values for LiC1, NaCl and KC1 at T , are 2.41,2.76, and 3.14 A, respectively. Calculated L* values for alkali chloride molecules in an air-acetylene flame are given in Table I. SMALL-PARTICLE VS LARGE-PARTICLE BEHAVIOR I t is important in applying the foregoing vaporization models to determine what constitutes "large-particle" and "small-particle" behavior. In typical flame or plasma spectrometry, the fog sent into the source consists mostly of droplets between 1 and 10 pm in diameter (41) and is formed from solution usually between 0.01% and 1% in total solids concentration. If the solid were NaCl of density 1.244 g/cm3, it can be calculated that the resulting dry aerosol will contain particles in the range 0.02 pm 5 r 5 1 pm. To determine whether these particles are "large" or "small", let us apply separately the heat-transfer-control and mass-transfer-control relationships. In mass-transfer-controlled vaporization (cf. eq 23), behavior changes from "large particle" to "small particle" when c u v r / ( l - cu/2)D = 1
(30)
Using the values for NaCl in Table I, one calculates that "small-particle" behavior should prevail for all particle radii below 19 pm if a = 0.2 and for all particles below r = 206 pm if a = 0.022! From this simple analysis, it would seem that the small-particle relation (eq 28) should apply to the vaporization of all NaCl particles in conventional flame or plasma spectrometry, if that vaporization were mass-transfer controlled. In fact, the vaporization times predicted by the general mass-transfer expression (eq 29) and the small-particle equation are not greatly dissimilar. If a = 0.01, eq 28 predicts a total time required for vaporization of a 1-pm-radius NaCl
ANALYTICAL CHEMISTRY, VOL. 59, NO. 24, DECEMBER 15, 1987
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2 3 2 5
2
1
1.5
Vaporization Coefficient
Time C m s )
