Anal.
chin. 1988, 60, 2653-2656
therm, the resistance to mass transfer, which is represented by the reverse of the mass transfer coefficient (i.e., l / m , acta as a smoothing factor and in the same time it couples the nonlinear effects. Therefore, when K increases, the selfsharpening effect becomes more intense, and the smoothing effect decreases (Figure 5). When K is large enough (Klarger than ca. 0.1-0.2), the chromatographicprocess itself becomes really operative and the stationary and mobile phases achieve near equilibrium. The shock layer forms for K larger than ca. 1 and becomes thinner and thinner. Then, with K becoming very large (i.e., K larger than ca. lo), the peak becoming narrower and taller, the velocity of the shock increases, the retention of the band maximum decreases, until the ideal Chromatography profie is reached,for an infiitely large value of K (in practice for K larger than ca. 50), and a zero value of the axial diffusion. There is a maximum in the value of the retention time of the band maximum, for values of the rate constant of a few tenth of a unit. Then, when K exceeds about 10, the retention time is nearly constant (15).
IV. Resolution between Bands and Column Efficiency. In linear chromatography, the resolution is defined by a Rayleigh equation, as the ratio of the distance between the maxima of the two bands to the average of their base width, i.e., twice the sum of their respective standard deviations
(33) This definition is absolutely unsuitable for applications in nonlinear chromatography, especially when shocks occur. The interactions between the components present in two bands which interfere (due to the competition between their molecules for adsorption sites on the adsorbent surface or, more generally, for interaction with the molecules or groups of the stationary phase) result in nonlinear effects. The only relevant parameter that can be used in this case is the yield of each of the mixture components at a stated degree of purity (18). This yield is an indication of the degree of interference between the bands. It is also directly related to the most important application of nonlinear chromatography, preparative separations, using overloaded elution.
2853
CONCLUSION
This work deals essentially with the problem of the elution profile of a single, pure compound band. Although important, this problem is not the most critical one in chromatography. Since this method is a separation method, we need to study the elution profiles of mixed bands, where two or more compounds struggle to separate from each other and interact, as soon as the experimental conditions (essentially the total concentration) are such that nonlinear behavior takes place. In a further contribution we shall study the interactions between the shock layers corresponding to the bands of two different compounds (19). LITERATURE CITED Aris, R.; Amundson, N. R. In “Madhemetical Methcds In Chemlcal En@neefhg;Prentice Hall: Englewood Cliffs, NJ, 1973 Vol. 2. Lax, P. D. Commun. Pure Appl. Math. 1957, IO, 537. De Vault, D. J . Am. Chem. Soc. 1943, 65, 532. Wilson, J. N. J . Am. Chem. Soc. 1940, 62. 1583. Rhee, H. K.; Aris, R.; Amundson, N. R. PMlOs. Trans. R . Soc. London, A 1970, 267, 419. Guiochon, 0.; Jacob, L. Chmatogr. Rev. 1971, 14, 77. Golshan-Shlrazl, S.; Gulochon, 0. Anal. Chem. 1966. 60. 2384. Rhee, H. K.; Amundson, N. R. Chem. Eng. Sci. 1974, 2 9 , 2049. Rouchon, P.; Schonauer, M.; Valentin, P.: Guiochon, G. In The Science of Chromatography; Bruner, F.. Ed.; Elsevier: Amsterdam, 1985; p 131. Golshan-Shirazi, S.: Guiochon, G.. In preparation. Rhee, H. K.; Amundson, N. R. Chem. Eng. Scl. 1074. 2 9 , 2049. Lln, BingChang; Wang, J b ; Lln, BingCheng J . Chromarogr. 1968, 438, 171. Lln. B.; Ma, 2.; Guiochon, G. in preparation. Whitham. 0. B. In Llnear and Non-//near Waves; Wiley: New York, 1974. Lln, B.; Golshan-Shirazi, S.; Guiochon, G., accepted for publication In J phvs. Chem Seinfeld, J. H.; LapMus, L. In Mathemetlcel M e w h Chemlcal M gineerlng; Prentice Hail: Englewood Cliffs, NJ, 1974; Vol. 3. OMdings, J. C. In Dynamb of Chromatography; Marcel Dekker, New York, 1984. Guiochon, G.; Ghcdbane, S. J . phvs. Chem. 1966, 92, 3682. Lin. B.; Golshan-Shirazi. S.; Ma, 2.; Guiochon, G. in preparation.
.
.
RECEIVED for review June 17,1988. Accepted August 31,1988. This work has been supported in part by Grant CHE8715211 of the National Science Foundation and by the cooperative agreement between the University of Tennessee and the Oak Ridge National Laboratory.
TECHNICAL NOTES Electron Mlcroscopy of Nanometer Partlcies in Freshwater Tsutomu Nomizu, Kenji Goto, and Atsushi Mizuike* Faculty of Engineering, Nagoya University, Chikusa-ku, Nagoya 464, Japan Nanometer particles, 1-100 nm, in freshwater play an important role to decrease the toxicity of heavy metals toward aquatic organisms by adsorption ( I ) . They are also known to contribute to sedimentation of heavy metals in estuarine regions (2). So far, however, little information has been available on the morphology and elemental compositon of individual particles. Scanning electron microscopy of particles smaller than 100 nm collected on a membrane fiiter does not give sufficient information, because of poor resolution in morphology and difficulty in X-ray microanalysis. An analytical electron microscope (AEM), a high-resolution transmission electron microscope (TEM) equipped with an X-ray microanalyzer, can be a powerful tool, although specimen 0003-270O/S6/0360-2653$0 1.50/0
preparation is more difficult. Direct or spray drying of the water sample on a film supported on a specimen grid is not ,applicable, because the concentration of nanometer particles is very low and dissolved salts crystallize out. Tipping et al. collected 0.1-0.5 pm hydrated iron(1II) oxide particles in lake water on a 5-nm-pore membrane filter by filtration, redispersed them in distilled water, and dried the suspension on a carbon film for observation and analysis with the AEM (3). Morphological modification as well as loss of particles may occur during the preparation, however. Biles and Emerson used centrifugation for collecting redispersed fiber aggregates, 0.5-2 pm, in water on a carbon film to observe fibers in beers with the TEM (4), but the specimen was not reproducible, 0 1988 American Chemical Society
2654 * ANALYTICAL CHEMISTRY, VOL. 60, NO. 23. DECEMBER 1, 1988
.,%;, , ., ,
,.:r*
Though part of particles smaller than 0.1 rm were retained on the filter (5), their number was estimated by observation under a scanning electron microscope to be less than 10% of that of the particles on a carbon film. The electron beam was focused on each particle, with the aid of scanning transmission electron microscopic observation, for 50 s for X-ray microanalysis with a take-off angle of 6 8 O . The elemental composition (in weight percent for elements heavier than sodium) was computed with an installed program based on Cliff and Lorimer's method (7). Electron diffraction data were not applicable to particles smaller than 0.1 rm.
-7"
RESULTS AND DISCUSSION
I
Flgu~e1. Monod spersed gold panicles collected by centr IJgallon on
a Caroon lim treated oy
glow
dscharge (conventional TEM image).
hecause a specimen grid was directly placed at the bottom of a centrifuge tube and agglomeration occurred during the preparation. Previously (5,6).we have developed a new specimen preparation terhnique by centrifugation using a specimen-grid holder, in which particles ranging from 0.1 to 1 rrm in freshwaters are dirertly snd reproducibly collected on a carbon film without agglomeration of particles. This note descrihes the usefulness of specimen preparation by centrifugation in the characterization of panicles smaller than 100 nm in pond and river waters using the AEM. Quantitative collection has been confirmed with monodispersed gold partirles, several nanometers in diameter.
EXPERIMENTAL SECTION The AEM used was a Hitachi H-EUU, operated at 100 kV, equipped with a Kevex 7OOU-Q energy dispersive X-ray microannlyzer. A Hitachi 65P-: ultracentrifuge was employed with an R P W swing-bucket mor to load three 5-mL rentrifirge tubes containing a specimen.grid holder. whore siEe was Smaller than that derrribed in a previous pnper (5)hernuse of the smaller size of the tube. A 1W20 nm thick rnrbun film on a 2Wmesh copper grid was treated hy glow diarharge at 5 mA for IS s in an Eiko IB-3 ion cnnter lo render the surface hydrophilic. A 4.mL water sample in each tube was centrifuged at .Woo0 rpm for 60 min. Freshwater samples were filtered rhruugh U.1-um-pore polycarhonate etzhec-track memhrnnt filters (Nuclepore)before centrifugation.
Flgure 2.
Adhesion of particles to a carbon film is essential to quantitatively collect particles in the proposed technique. In the case of weak adhesion, the particles are detached from the film and lost by diffusion after the termination of centrifugation. Displacement of spherical particles in water by the Brownian motion is inversely proportional to the square root of the particle diameter, e.g., ca. 1 pm in a second for 1O-nmspherical particles, theoretically (8). when the adhesion is weak, some of the particles may also be lost during removal of water or may aggregate on the carbon film during drying. T o evaluate the collection of nanometer particles on the carbon f h ,a monodispersed gold suspension was used, which was prepared by mixing 90 mL of 0.11 fig of Au/mL gold(II1) chloride solution with 10 mL of 0.2 mM sodium citrate (9). The prepared suspension wax directly sedimented on a carbon film by centrifugation without preliminary filtration through a 0.l-rm-pore filter. The particles shown in Figure 1have an arithmetic mean diameter, Z:n;di/N(n;is the number of particles having a diameter of d;, measured on the electron micrograph; N is the total number of particles), of 5.9 nm with a standard deviation of 1.8 nm and a volumetric mean diameter, (Z: nid:/N)'l3, of 6.5 nm. According to the calculation based on Stokes' law, spherical gold particles larger than 2.5 nm are quantitatively sedimented by centrifugation under the prescribed conditions. If gold particles having a diameter of 6.5 nm in the suspension (0.1 pg of Au/mL) are quantitatively collected, 1.7 X lo3 particles/pm* should he counted on the carbon film. Actually 1.5 X lo3 particles/pm2 were counted; i.e., ahout 90% of gold particles was collected. The treatment of the carbon film by glow discharge was effective for a few days, but adhesion of the particles was remarkably deteriorated on the film 1 week after the treatment. Density difference between gold (19.3 g cm-3) and colloidal particles in freshwaters (ca.2 g cm4) does not affect diffusion phenomena of nanometer particles, although a difference in adhesion to
Conventional TEM images of 10-100-nm particles in freshwaters: (a)pond water (%e Kagamigake)and (b) river water (the Shonaigawa).
ANALYTICAL CHEMISTRY, VOL. 60. NO. 23, DECEMBER 1, 1988
2655
a
10
.3
s
5
Flgure 4.
X-ray spectrum of particle no. 21 in
Figure 2a.
Particle No.
b 100-
s- . 3
50.
-
nFIgure 3.
5 Particle
10
15
No.
Elemental composfiions(forelements heavier than Na) of
10-100-nm particles in freshwaters(arranged in order of irm content): (a)pond water (the Kagamigalke)and (b) river water (the Shonaigawa).
the film between the two is uncertain. Thus particles larger than 10 nm in freshwaters can be collected nearly quantitatively under the prescribed conditions. Another surface treatment, immersion of the carbon film in a cationic surfactant solution (0.03% dkteryldimethylammoniwn chloride) or an anionic surfactant solution (0.01% sodium oleate) to render the surface positively or negatively charged, was not effective. Figure 2 shows the TEM images of 10-100 nm particles in freshwater. The elemental compositions (for elements heavier than sodium) of the particles numbered in Figure 2 are given in Figure 3 by using the same number. In the X-ray microanalysis of nanometer particles, the contamination of silicon in the microscope (probably from silicone grease) caused errors of &lo% in elemental composition.
In both samples, most particles consist of iron, silicon, and aluminum. These particles seem to be aggregates consisting of hydrated iron(II1) oxide and silica, of hydrated iron(II1) oxide and clay (no. 27 in Figure 2a and no. 15 and 18 in Figure 2b) and of the above three, their morphology and elemental compositions being compared with those of particles in the 0.1-1-pm range (6). Predominant particles in the l(t100 nm range are aggregates consisting of hydrated iron(II1) oxide and silica, compared with clay associated with hydrated iron(II1) oxide in the 0.1-1 pm range. Phosphorus and calcium were detected in some particles, having low-contrast TEM images (no. 25 in Figure 2a and no. 14 in Figure 2b). They seem to be associated with organic or biological matter, because strong signals of phosphorus and calcium were detected in microorganisms in the 0.1-l-pm range (6).Threadlike particles (indicated by arrows in Figure Za), which give weak X-ray signals of iron only, are also presumed to be organic or biological. The X-ray spectrum of particle no. 21, ca. 20 nm, in Figure 2a is shown in Figure 4. The relatively high background wm not caused by bremstrahlung from the observed particle but mainly by X-ray due to the interaction of undetected X-ray with the apertures, lenses, shielding materials, etc. in the electron microscope. The error caused by the background, however, was estimated to be less than 5% in elemental composition for particles larger than ca. 40 nm. The low signal-to-background ratio, fluctuation of electron beam spot (caused by deformation of the carbon film or backlash of the specimen handling mechanism) and silicon contamination practically limit the minimum particle size to about 10 nm in X-ray microanalysis, although morphological studies can be done for much smaller particles. The proposed technique can also be applied to electron microscopic characterization of nanometer particles in various solutions, especially those containing salts, such as seawater and wastewater. ACKNOWLEDGMENT The authors thank Professor K. Takata, Radioisotope Center, for providing various apparatus, and Mr. T. Hanaichi and MI. T. Iwamoto, Electron Microscope Center, for help and advice in the AEM experiment. Registry No. Fe, 7439-89-6; AI, 7429-90-5; Si, 7440-21-3; P, 7723-14-0; Ca, 7440-70-2; H,O, 7732-18-5.
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ANALYTICAL CHEMISTRY, VOL. 60, NO. 23, DECEMBER 1, 1988
LITERATURE CITED (1) Florence, T. M.; Batby. 0. E. CRC Cr#. Rev. Anal. Chem. 1980, 9 , 219. (2) Hart, 8. T.; Davles, S. H. R. Estuarine, Coastal SheH Sci. 1981, 12, 353. (3) . . TiDDlnG. E.: Woof, C.: Cooke. D. Geochlm. C o s m h / m . Acta 1981, 45: 1411-1419. (4) Bibs. 8.; Emerson, T. R. Nature (London) 1968. 219, 93. ( 5 ) Nomizu, T.; Mizulke. A. Mkrochim. Acta 1086, 1986 I , 65-72.
(6) Nomizu, T.; Nozue, T.; Mlzuike, A. Mlkrochim. Acta 1988. 7987 11, 99-106. (7) Cliff, 0.;Lorimer, G. W. J . Microsc. lW5, 103, 203-207. (8) Shaw. D. J. Introduction to CdlOM and Surface Chemistry, 2nd ed.; Butterworuls: London, 1970; p 20. (9) Takiyama, K. Bull. Chem. Soc. Jpn. 1958, 37,944-950.
RECEIVED for review May 27,1988.Accepted August 15,1988.
CORRECTION Error Estimation Using the Sequential Simplex Method in Nonlinear Least Squares Analysis Gregory R. Phillips and Edward M. Eyring (Anal. Chem. 1988,60,738-741). Table I contains two misprints, which affect the validity of the equations in the text. A corrected version of Table I is given below:
H vector: B matrix: Q matrix:
= 2xOt - (xi2 + 3 x O 2 ) / 2 bii = 2(x? + xo2 - 2xoi2) bij = ?(xi?_+ xo2- xo? - x o f ) qij = 81.8. . - 8I,O .
ai
i = 1, ..., k i = 1, ..., k
+
i j i = 1, ..., k ,
j = 1, ..., k
Additionally, eq 6 should be
iitB-lB-% < 1 instead of
iitB-lB-l1 c 1/4 Finally, eq 7 should be E
= s2QB-'Qt
instead of e = s22QB-'Qt
The estimated errors for the simplex method in Table I1 should be reduced by the square root of 2,bringing them into closer agreement with the errors calculated using the Marquardt and Gauss-Newton methods.
