Fundamental Plasma Investigations: Plasma Simulation

Julie A. Horner

To complement experimental results obtained through fundamental study of the inductively coupled plasma (ICP), the Hieftje group has recently undertaken theoretical modeling of the ICP. It is the goal of the theoretical subgroup to gain a fundamental understanding of the processes occurring in the ICP. This is addressed by first developing a model for the processes of interest and second by comparison of results of the model with experimental findings. Agreement between experiment and theory implies achievement of the original goal. The theory may then be used to explain such complex effects as those of added substances on analyte signal levels. Further, if theory and experiment agree well, the theory alone might be used, for example, to develop an ICP optimized for analytical performance. This optimization via simulation could include design of an ICP torch, followed by determination of flow rates and powers to be used with the torch. Furthermore, simulation could be used to determine which transitions are likely to be most intense for the set of optimized conditions. Finally, an accurate computer model could be used to predict results for experiments which would be difficult, dangerous or impossible to perform.

There are three areas in ICP emission spectrometry for which it would be beneficial to provide accurate simulation. The first is the generation of the physical environment which establishes the bulk properties of the plasma. The second is a group of processes which result in the transformation of analyte aerosol into atomic vapor; i.e. desolvation, vaporization and atomization. The third is generation of observed analyte emission signals, i.e. excitation and ionization of analyte atoms.

There are two steps involved in the computer simulation. The first is development of a suitable set of mathematical equations for description of all relevant processes. The second is the development of an algorithm for solution of the equations. The simulation of each of the three areas mentioned above is discussed in further detail below.

1. ICP Model: Calculation of ICP Fundamental Properties

Simulation of the fundamental properties of the ICP (electron temperature, electron number density and gas temperature) involves mathematically modeling how energy is coupled into and transported within the plasma medium. For a spectrochemical ICP, these processes are described by equations for an external electromagnetic field, and for velocity, momentum, and energy of all species in the discharge. All of these equations can be cast in a similar form so a single method for their simultaneous solution can be used. The method for solving the equations, known as the SIMPLER algorithm [1], is used in the code developed by P. Yang, R. M. Barnes, J. Mostaghimi, and M. I. Boulos [2] for the simulation of analytical plasma properties. In addition to the processes which are included in most models for the ICP, their software has been recently modified in this laboratory by P. Yang to include several unique features of an analytical ICP. First, all properties, including the EM field, are calculated for two spatial dimensions because it has been shown experimentally that the ICP is spatially inhomogeneous. Second, the diffusion of air into the ICP has been included because this has been shown to play a large role in defining the outer edge of the plasma. Finally, provision has been made for the modeling of water and analyte vapor diffusion throughout the ICP.

A recent comparison of results for a dry argon ICP from this model with experimental results obtained using Thomson and Rayleigh scattering methods shows agreement to within 20% for most areas of the plasma. In particular, the electron temperature and electron number density show very good agreement (10%) with experiment. The gas temperature shows fairly good agreement but has an apparent systematic error of approximately +1000K. Modifications to the model and examination of data collection and processing methods are underway to determine the cause of the discrepancy.

2. Aerosol-Diffusion Model: Simulation of Analyte Number Densities

Desolvation, vaporization, atomization and partial ionization of the aerosol introduced into the ICP are governed by the physical environment of the ICP as well as by the kinetics of the processes themselves. Thus, the model for fundamental parameters (described above) must be used in conjunction with a kinetic model to give a time-resolved description of the passage of a droplet/particle through the ICP. The dynamic processes of desolvation, vaporization and atomization proceed at a rate which will be limited by one of three transport processes: the transport of heat to the surface of the droplet or particle (heat transfer); the transport of droplet/particle material away from the surface of the droplet or particle (mass transfer); or diffusion of species out of the stagnant layer surrounding the droplet or particle (diffusion). The form of the equation describing the desolvation or vaporization process depends on which transport process governs the kinetics. Once the appropriate equation for the droplet or particle trajectory has been defined, a droplet/particle history can be calculated for the desired ICP conditions.

The droplet/particle model currently describes the loss of solvent (desolvation) and the subsequent vaporization of a solute particle, atomization of the solute vapor and partial ionization of the resulting analyte atoms. Single aerosol-droplet trajectories as well as multi-aerosol droplet profiles can be simulated for a variety of ICP operating conditions. Separate calculations have been carried out for pure solvent droplets, for pure solute particles, and for aerosol droplets containing one or a mixture of two solutes. The results of these studies are outlined below.

The effect of droplet size, ICP central-channel flow rate and applied rf power on single- droplet trajectories has been determined. The time (tcd) or height (hcd) of complete desolvation of a droplet is quadratically related to its diameter. This is because the desolvation process is controlled by heat transfer. In addition, the tcd is related roughly linearly to the central channel flow rate, and inversely to the applied rf power. These dependences reflect the changes in gas temperature that occur with changing flow rate and applied power. Multi-droplet profiles have been calculated for droplet distributions from several commercial nebulizer/spray-chamber combinations. These results indicate that, in the absence of any effects of the droplets on plasma parameters, the aerosol from a typical sample introduction system is almost completely desolvated at 30 mm above the load coil (ALC). Droplet histories obtained using this droplet model have recently been compared with those obtained by experiment. Experiments show that 30 µm diameter droplets are completely desolvated at a height of 18 mm ALC, whereas the theory indicates that a 20 µm droplet is desolvated at 18 mm ALC. Work is underway to determine the origin of the difference in results.

The effect of particle size, ICP central-channel flow rate and applied rf power on single- particle trajectories has also been determined for several solutes. The time (tcv) or height (hcv) of complete vaporization is linearly related to the particle diameter. The hcv is not related in any obvious way to the thermal properties of the solute, however, it is related to the central channel flow rate and applied power in the same fashion as was the hcd for droplet desolvation.

The changes in particle vaporization that occur on addition of a second solute to the sample have been examined using the same desolvation and vaporization model. Properties of the mixed solute are calculated from the individual solute properties and from the mole fractions of each solute. In order to compare directly with experimental results, radially resolved profiles have been calculated by including diffusion and flow of the analyte away from the central channel. With these inclusions, analyte number densities as a function of height and radial distance in the plasma have been simulated and compared to those obtained by experiment. Absolute values of number density obtained from the simulation are similar to those obtained by experiment; however simulated ion to atom number density ratios are in general lower than experimental values. Good correlation between simulated and experimentally observed changes in analyte densities as a result of the added element was obtained for Ca in the presence of Zn, Li and Ba. For some of the other mixed solute pairs (Ca with Na, Ag and K), however, there are notable differences between simulation and experiment. There are several possible reasons for the differences. First, the simulation package currently assumes that no changes in plasma properties take place on addition of an aerosol; experiments have been carried out which are in disagreement with this assumption. Second, the properties of the mixed solute may not be well represented by a linear combination of the individual properties weighted by their mole fractions. And, third, atomization and ionization are assumed to be the only chemical changes taking place for the analyte solute. Further, these processes are assumed to be at equilibrium and to be instantaneous, whereas they are likely under kinetic control. Any or all of these assumptions may be incorrect and could result in differences between simulated and experimental analyte number density profiles.

3. Emission Model: Simulation of Analyte Emission Signals.

Excitation and ionization of atomic vapor produced by the combined processes of desolvation, vaporization and atomization are also kinetic in nature. A well used model describing these processes, first introduced by R. Lovett in 1982, is the Collisional- Radiative (CR) model. Population of excited states is proposed to be limited by the rate of collisions between particles and electrons while depopulation of excited states is suggested to be governed by three competing processes: radiative loss, collisional deexcitation, or further excitation/ionization. A CR model for the description of excitation and ionization of calcium has been developed and has been used in conjunction with analyte diffusion profiles from the fundamental model for generation of spatial emission profiles of analyte species in the ICP.

The ultimate goal of the ICP simulation work in this group is to develop a single model which accurately describes all of the three areas of ICP emission for any desired set of operating conditions. In order to achieve this goal, several changes must be made to the existing model. First, the aerosol-diffusion model must take into account radial movement of the analyte due to convection, i.e. forced movement. Second, the analyte excitation/ionization model must be expanded to include processes involving the added element. The complete aerosol model should then be integrated with the fundamental model so that reciprocal interactions could be examined. Reciprocal interactions refer to the effect of droplet desolvation, particle vaporization, atomization, ionization and excitation on the plasma. This software could be used to simulate, for example, spatial analyte emission profiles under a variety of operating conditions and in the presence of a variety of concomitant species. The simulated profiles could then be compared with experimental profiles to gain an improved fundamental understanding of the ICP.

[1] S. V. Patankar, "Numerical Heat Transfer and Fluid Flow",Hemisphere Publishing Corporation, New York, NY.

[2] P. Yang, R. M. Barnes, J. Mostaghimi and M. I. Boulos, Spectrochim. Acta Part B, 44B, 657-666 (1989).

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