Anal. Chem. 1991. 63.2727-2734
devices to the determination of a single acidic component in a gas mixture. Proton-conductingmembranes, such as Ndion, seem to be more suitable for this purpose than anion-exchange membranes. Long-term stability of the gold- or platinumcovered Nafion electrodes is satisfactory, as no appreciable change of their conductance is observed after several weeks of continuous use. This stability is attributed to the solid-state structure of the probe. This prevents slow detachment of particles of conductive material from the electrode surface as observed from SPE systems used in liquid media (12). Moreover, stability toward corrosive gases such as HC1 and SOz can be related to chemical inertness of the gas-sensing element (consisting of membrane and electrodes). Although no selectivity has been sought in the present study, lack of conductance response of the metal/SPE toward gases such as Nz, 02, and COz enables possible use of these devices for quantitative measurements of ambient and industrial SO2 pollutant. This method seems to be preferable to one employing liquid melts (30) or high-temperature solid-state devices (31). Future work will be focused upon investigation of this use as well as on geometry for greater sensitivity and more rapid response time. Registry No. Au, 7440-57-5; Pt, 7440-06-4; SOz, 7446-09-5; HC1, 7647-01-0 Nafion 117, 66796-30-3.
LITERATURE CITED N M , M.; Kanefusa. S.; Haradome, M. J . Elecfrochem. SOC. 1978, 125, 1676-1679. Wimdischmann, H.; Mark, P. J . Electrochem. Soc. 1979, 726, 627-633. Duh, J. 0.;Jou, J. W. J . Ekhochem. Soc. 1989, 136, 2740-2747. Appbby, A. J.; Yeager, E. 6. €nergy(Oxford) 1986, 11. 137-152 and references cited therein. Lu, P. W. T.; Srinivasan, S. J . Appl. Elecfrochem. 1979, 9 , 269-263. Fujlta, Y.; Nakamura, H.; Muto, T. J . Appl. Electrochem. 1986, 76, 935-940.
2727
Enea, 0. J . Ektroanal. Chem. InterfaclelEkfrochem. 1987, 235, 393-401. Ogumi, 2.; Inaba, M.; Ohashi, S.; Uchida. M.; Takehara, Z. f h t r o chim. Acta 1988, 33, 365-370. Kita, H.; Nakajima, H. Electrochim. Acta 1986, 31, 193-200. De Wulf, D. W.; Bard, A. J. J . Electrochem. SOC. 1988. 735. 1977-1965. Kaaret, T. W.; Evans, D. H. Anal. Chem. 1988, 60, 657-662. Schavon. 0.;Zoni. G.; Bontempelli, 0. Anal. Chlm. Acfa 1989, 227, 27-4 1. Schiavon, 0.;Zoni, G.; Bontempelii, 0.;Farnia, G.; Sandona, G. Anal. Chem. 1990. 62, 293-298. Katayama-Aramata, A.; Nakalima, ti.; Fujikawa. K.; Kita. H. Elechochlm. Acta 1983, 28, 777-780. Katayama-Aramata, A.; Ohnishi, R. J . Am. Chem. Soc. 1983, 705, 658-659. Tables of Wavenumbers for the Calibration of Infra-Red Specfromefers; IUPAC; Butterworth: London, 1961; pp 654-657. Cai. 2.; Liu. C.; Martin, C. R. J . Elecfrochem. SOC. 1989. 736, 3356-3361. &ot, W. G. F.; Munn, G. E.; Walmsley, P. N. J . Electrochem. Soc. 1972. 179, 106C-111'2. Yeo. S. C.; Eisenberg, A. J . Appl. folym. Sci. 1977, 27, 675-890. Yeo. R. S. J . Electrochem. Soc. 1983, 730, 533-536. Hsu, W. Y.; Gierke, T. D. J . Membr. Sci. 1983, 13, 307-326. Yeager, H. L.; Kipiing, B. J . phvs. Chem. 1979. 83. 1836-1839. Sakai, T.; Takenaka, H.; Wakabayashi. N.; Kawami, Y.; Torikai, E. J . Electrochem. Soc. 1985. 732, 1328-1332. Verbrugge. M. W.; Hill, R. F. J . Electrochem. SOC. 1990, 737, 886-893. Verbrugge, M. W.; Hill, R. F. J . Electrochem. Soc. 1990, 737, 893-699. Verbrugge, M. W.; Hill, R. F. J . Electrochem. Soc. 1990, 737, 113 1-1 138. Randin, J. P. J. Elecfrochem. SOC.1982, 729, 1215-1220. Harth. R.; Mor, U.; Ozer, D.; Belteiheim. A. J . Elecfrochem. Soc. 1989, 736, 3863-3667. Martin. C. R.; Doilard, A. J. Electroanal. Chem. Interfacial E k f r o chem. 1983, 759, 127-135. Saizano, J.; Newman, L. J . Electrochem. SOC. 1972, 779, 1273-1278. Jacob, K. T.; Bhogeswara, R. J . Elecfrochem. Soc. 1979. 126, 1842-1847.
RECEIVED for review December 18, 1990. Accepted August 23, 1991.
Characterization of Sample Heterogeneity in Secondary Ion Mass Spectrometry by the Use of a Sampling Constant Model Frank P. L. Michiels and Freddy C. V. Adams*
Department of Chemistry, University of Antwerp (UIA), Universiteitsplein 1, B-2610 Wilrijk, Belgium David S . Bright and David S. Simons
Center for Analytical Chemistry, National Institute of Standards and Technology, Gaithersburg, Maryland 20899
An hdepth study was undertaken to evaluate the appllcablllty of the sampling constant concept for the characterlzatlon of hetemgemhy of ekfnentai distributions In ion mkroscopy. A new model, whlch take8 Into account the Intensity spread of inclusions, Is described. The model is tested on low alloy steel and brass standard reference samples, relylng on Image registratbn wlth a reslstive anode encoder and Image analysls. The agreement between the model and the experhnentai results Is satlsfactory.
INTRODUCTION Despite its marked microanalytical potential, secondary ion mass spectrometry (SIMS) is very often used for the bulk 0003-2700/91/0363-2727$02.50/0
analysis of solids. Commonly, standard samples that have been certified for their bulk composition are used as Calibration samples in quantitative analysis. Theoretical and semitheoretical calibration approaches are mostly evaluated using bulk materials. In both these instances, it is often implicitly assumed that the sample is homogeneous, i.e., that there are no spatial differences in the concentration of the constituents. However, the composition of the sampled subvolume is not necessarily representative for the overall composition of the sample. Due to the very small sampled volume in SIMS, large errors often arise. This problem has been recognized before and some efforts to overcome it have been undertaken (1-6). Since heterogeneity effects are often very important in the overall quantification error, some understanding of the sampling problem is essential to any com0 1991 American Chemical Society
2728
ANALYTICAL CHEMISTRY, VOL. 63, NO. 23, DECEMBER 1, 1991
plete evaluation of quantification attempts in SIMS. By far the most important work on the sampling error in SIMS has been performed by Morrison and co-workers (I, 7, 8). The basic idea of the work is that the heterogeneity of the lateral distribution of an impurity element in a solid sample can be quantitatively described by a sampling constant. This sampling constant can be used to estimate the heterogeneity error for a certain analysis or to estimate the number of analyses necessary to obtain a certain precision. In the original work, the value of the sampling constant is only dependent on the number of inclusions per unit area (the density) that contain the impurity element of interest. A spread in the intensities of the inclusions is assumed to be absent or only of secondary importance. In experimental practice, though, it is frequently observed that inclusions have a considerable spread in intensity. Furthermore, the experimental determination of sampling constants often yields unsatisfactory results. Therefore, a further elaboration of the sampling constant concept seems necessary. In this article, the basics of the sample constant concept will be reexamined, using ion microscopy as a tool for the measurement of heterogeneity. A new model to describe the heterogeneity is proposed and verified experimentally. The new model contains only few constraints and it reveals that the concept of the sampling constant concept remains valid, even if inclusions with widely varying intensities are involved, or when they cover a relatively large surface area. In a companion article, difficulties encountered in the experimental determination of sampling constants will be discussed. Apart from conventional errors such as instrumental errors, there appears to be a large contribution from statistical errors. Computer simulations allow us to model these types of errors. A companion article (9) will elaborate on the limitations and possible errors sources of the model. THEORY Description of t h e Extended Model. To describe the heterogeneity of a sample for the extended model, the following assumptions are proposed. (i) The element of interest is present in two phases in the solid material. The first phase consists of a homogeneously distributed fraction of the element, the concentration of which is the same over the whole sample volume. In practice, this concentration may, for instance, correspond to the solid solubility limit of the impurity in the matrix. The second phase consists of inclusions. These are small precipitates that contain a higher concentration of some or all of the impurity elements than the matrix. (ii) The inclusions are considered to be randomly distributed throughout the sample volume, and it is assumed that there are no spatial correlations between inclusions of different sizes or intensities. The intensity of an inclusion is the secondary ion intensity ( ~ integrated 4) over the whole area of the inclusion, for the element under consideration, measured in the specified analytical conditions. Although this may not always be the case, e.g., when the impurity elements are precipitated at grain boundaries, it seems to be largely valid. (iii) The model assumes that the ion intensity map is completely equivalent to the concentration map, Le. that matrix effects do not occur within the sample volume analyzed. (iv) If this condition is fulfilled, the values of the sampling constant will be independent of the analytical technique used and can be used to predict sample heterogeneity as measured with SIMS as well as other analytical techniques (provided they comply with the restrictions listed here). (v) In analytical practice, this condition is hard to meet, especially in SIMS where large matrix effects occur. In this case, the use of the sampling constant is restricted to the description of point to point variations of the intensity, rather
m
Figure 1. Threedimensional histogram showing the (hypothetical)size and intensity distribution of inclusions in a solid matrix.
than of the actual concentration. In our experiments, matrix effects were minimized by using oxygen bombardment. (vi) The SIMS analysis is essentially two-dimensional. Since the depth sputtered during the registration of an ion image is usually several orders of magnitude smaller than the lateral dimension, this condition is rarely a restriction of the model. It means that the sample will be characterized in terms of analyzed area, surface area of the inclusions, etc., rather than in terms of volumes. (vii) The variation of the measured ion currents can be attributed completely to heterogeneity effects. In practice, this will be realized by isolating the contribution of the heterogeneity and the contribution of other error sources (instrumental errors, counting statistics). (viii) The inclusions present in the sample are not necessarily identical. Variations in intensity as well as in size of the different inclusions must be allowed. The distributions can be classified according to size as well as intensity. This is the main difference with the conventional model, where the inclusions were assumed to have similar intensities. (ix) The size of the inclusions needs to be small compared to the size of the analyzed area. Mathematical Derivation of the Sampling Constant. Figure 1shows a hypothetical intensity and size distribution of the inclusions present in a sample. The inclusions are classified into m different surface area classes and n different intensity classes. Ak is the surface area of an inclusion in the kth surface area class, and If'is the intensity of one inclusion in the lth intensity class. The total secondary ion current (Ibt)in any subarea A,, of the sample, will be given by I n "
ztot.= I , + C CN,,,I~ k=ll=l
where Nk,, is the number of inclusions of the kth surface area class and the lth intensity class in the area of analysis, and I, is the contribution of the homogeneously distributed fraction of the element. If we assume that there are no correlations between the different types of inclusions, then we can write the variation of the total intensity as m
n
since the contribution of the homogeneously distributed fraction (I,) is constant. The variations of the total intensity are caused by variations of the number of inclusions of each type in different areas of analysis. The variation in the number of inclusions of each type can be determined if we assume that the inclusions are uniformly randomly distributed. We can divide the sample surface into
ANALYTICAL CHEMISTRY, VOL. 63, NO. 23, DECEMBER 1. 1991
'unit cells" of area Ah, which is the surface area of one inclusion. If A. is the size of the area of analysis, then the number of unit cells of the kth class (nk) within the area of analysis can be written as nk
= AO/Ak
(3)
The probability that a random unit cell contains an inclusion of the kth size class and the lth intensity class ( p k , l ) is then Pk,l
dk.lAk
(4)
where dk,lis the area density of inclusions of the kth surface area class and the lth intensity class. Assuming a random distribution of the inclusions, we can write for the population standard deviation (binomial distribution of the number of inclusions) bhrh,?
- Akdk,l)
AOdk,l(l
(5)
2729
Equation 12 predicts that the error due to heterogeneity decreases as the size of the area increases, as would be intuitively expected. In a physical sense, the sampling constant describes the heterogeneity of an element in a certain matrix (sample) and may therefore be considered a sample characteristic (for every element). Equation 12 can be used to estimate the minimum size of the area of analysis, when the error due to heterogeneity is given. Moreover, it allows the estimation of the number of replicate analyses, necessary to attain a certain precision (I). The sampling constant has the dimension of distance. This is the most general definition of the sampling constant. Depending on the contribution of the different terms, simplifications are sometimes allowed. Three main simplifications will be used and can be described as follows. Small Area Coverage. If the area coverage is well below unity (pk,l
